diff --git "a/KdFRT4oBgHgl3EQf0zhG/content/tmp_files/load_file.txt" "b/KdFRT4oBgHgl3EQf0zhG/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/KdFRT4oBgHgl3EQf0zhG/content/tmp_files/load_file.txt" @@ -0,0 +1,1730 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf,len=1729 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='13654v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='GT] 31 Jan 2023 MULTI-AGENT CONTRACT DESIGN: HOW TO COMMISSION MULTIPLE AGENTS WITH INDIVIDUAL OUTCOMES ARXIV PREPRINT Matteo Castiglioni Politecnico di Milano matteo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='castiglioni@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='it Alberto Marchesi Politecnico di Milano alberto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='marchesi@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='it Nicola Gatti Politecnico di Milano nicola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='gatti@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='it February 1, 2023 ABSTRACT We study hidden-action principal-agent problems with multiple agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' These are problems in which a principal commits to an outcome-dependent payment scheme (called contract) in order to incen- tivize some agents to take costly, unobservable actions that lead to favorable outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Previous works on multi-agent problems study models where the principal observes a single outcome deter- mined by the actions of all the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Such models considerably limit the contracting power of the principal, since payments can only depend on the joint result of all the agents’ actions, and there is no way of paying each agent for their individual result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In this paper, we consider a model in which each agent determines their own individual outcome as an effect of their action only, the principal observes all the individual outcomes separately, and they perceive a reward that jointly depends on all these outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This considerably enhances the principal’s contracting capabilities, by allowing them to pay each agent on the basis of their individual result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We analyze the computational com- plexity of finding principal-optimal contracts, revolving around two newly-introduced properties of principal’s rewards, which we call IR-supermodularity and DR-submodularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Intuitively, the for- mer captures settings with increasing returns, where the rewards grow faster as the agents’ effort increases, while the latter models the case of diminishing returns, in which rewards grow slower instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' These two properties naturally model two common real-world phenomena, namely disec- onomies and economies of scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In this paper, we first address basic instances in which the principal knows everything about the agents, and, then, more general Bayesian instances where each agent has their own private type determining their features, such as action costs and how actions stochastically determine individual outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' As a preliminary result, we show that finding an optimal contract in a non-Bayesian instance can be reduced in polynomial time to a suitably-defined maximization problem over a matroid having a particular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Such a reduction is needed to prove our main positive results in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We start by analyzing non-Bayesian instances with IR- supermodular rewards, where we prove that the problem of computing a principal-optimal contract is inapproximable in general, but it becomes polynomial-time solvable under some mild regularity assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, we study non-Bayesian instances with DR-submodular rewards, showing that the problem is inapproximable also in this setting, but it admits a polynomial-time approximation algorithm which outputs contracts providing a multiplicative approximation (1 − 1/e) of the prin- cipal’s reward in an optimal contract, up to a small additive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In conclusion, we extend our positive results to Bayesian instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' First, we provide a characterization of the principal’s opti- mization problem, by showing that it can be approximately solved by means of a linear formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This is non-trivial, since in general the problem may not admit a maximum, but only a supremum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, based on such a linear formulation, we provide a polynomial-time approximation algorithm that employs an ad hoc implementation of the ellipsoid method using an approximate separation oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We prove that such an oracle can be implemented in polynomial time by exploiting our posi- tive results on non-Bayesian instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Surprisingly, this allows us to (almost) match the guarantees obtained for non-Bayesian instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ARXIV PREPRINT - FEBRUARY 1, 2023 1 Introduction Over the last few years, principal-agent problems have received a growing attention from the economics and compu- tation community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' These problems model scenarios in which a principal interacts with one or more agents, with the latter playing actions that induce externalities on the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We focus on hidden-action problems, where the princi- pal only observes some stochastically-determined outcome of the actions selected by the agents, but not the actions themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The principal gets a reward associated with the realized outcome, while an agent incurs in a cost when performing an action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Thus, the principal’s goal is to incentivize agents to undertake actions which result in profitable outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This is accomplished by committing to a contract, which is a payment scheme defining how much the principal pays each agent depending on the realized outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The classical textbook example motivating the study of hidden-action principal-agent problems is that of a firm (prin- cipal) hiring a salesperson (agent) in order to sell some products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The salesperson has to decide on the level of effort (action) to put in selling products, while the firm only observes the number of products that are actually sold (out- come).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In such a scenario, it is natural that the firm commits to pay a commission to the salesperson by stipulating a contract with them, and that such a commission only depends on the number of products being sold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Nowadays, the study of principal-agent problems is also motivated by the fact that they are ubiquitous in sev- eral real-world settings, such as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', crowdsourcing platforms (Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2016), blockchain-based smart con- tracts (Cong and He, 2019), and healthcare (Bastani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The computational aspects of principal-agent problems with a single agent have been widely investigated in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Instead, only few works study problems with multiple agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Some notable examples are the papers by Babaioff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2006) and Emek and Feldman (2012), and the very recent preprint by Duetting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' These works address models where the principal observes a single outcome determined by the actions of all the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Such models considerably limit the contracting power of the principal, since payments can only depend on the joint result of all the agents’ actions, and there is no way of paying each agent for their individual result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In this paper, we introduce and study principal-agent problems with multiple agents—compactly referred to as principal-multi-agent problems—in which each agent determines their own individual outcome as an effect of their action only, the principal observes all the individual outcomes separately, and they perceive a reward that jointly de- pends on all these outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Our model fits many real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For instance, in settings where a firm wants to hire multiple salespersons, it is natural that the firm can observe the number of products being sold by each of them individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Additionally, as we show in this paper, our model also allows to circumvent the equilibrium-selection issues raised by the problems studied in (Babaioff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Emek and Feldman, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Duetting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In- deed, as we discuss later in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='2, such issues originate from the appearance of externalities among the agents, which are instead not present in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='1 Original Contributions We investigate the computational complexity of finding optimal contracts in our principal-multi-agent problems with agents’ individual outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Our analysis revolves around two properties of principal’s rewards, which we call IR- supermodularity and DR-submodularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Intuitively, the former captures settings with increasing returns, where the rewards grow faster as the agents’ effort increases, while the latter models the case of diminishing returns, in which rewards grow slower as the effort increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' These two properties naturally model two common real-world phenomena, namely diseconomies and economies of scale, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In the first sections of the paper (namely Sections 2, 3, 4, and 5), we study basic principal-multi-agent problems in which the principal knows everything about agents, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', their action costs and the probability distributions that their actions induce over (individual) outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, in Section 6, we switch the attention to the far more general Bayesian settings in which each agent’s action costs and probability distributions depend on a private agent’s type, which is unknown to the principal, but randomly drawn according to a commonly-known probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' After introducing, in Section 2, all the preliminary concepts related to the non-Bayesian version of our principal-multi- agent problems, in Section 3 we provide a useful preliminary result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We show that the problem of computing an optimal contract in a non-Bayesian instance can be reduced in polynomial time to the maximization of a suitably-defined set function over a matroid having a particular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Specifically, we call the matroids introduced by our reduction 1-partition matroids, since their ground sets are partitioned into some classes and their independent sets are all the subsets which contain at most one element for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' At the end of the section (more precisely in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='3), we also provide an additional preliminary result, by showing that there exists a polynomial-time algorithm for maximizing particular set functions, which we call ordered-supermodular functions, over 1-partition matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This will be useful to derive our positive result in the following Section 4, and it may also be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2 ARXIV PREPRINT - FEBRUARY 1, 2023 In Section 4, we provide our main results on non-Bayesian instances with IR-supermodular principal’s rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We start with a negative result: for any ρ > 0, it is NP-hard to design a contract providing a multiplicative approximation ρ of the principal’s expected utility in an optimal contract, even when both the number of agents’ actions and the number of outcomes are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, we show how to circumvent such a negative result by introducing a mild regularity assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Specifically, we prove that, in instances with IR-supermodular principal’s rewards that additionally satisfy a particular first-order stochastic dominance (FOSD) condition, an (exact) optimal contract can be found in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This is accomplished by exploiting the reduction introduced in Section 3, and by proving that, for such instances, the resulting set function is ordered-supermodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In Section 5, we switch our attention to non-Bayesian instances with DR-submodular principal’s rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Similarly to the preceding section, we start with a negative result: for any α > 0, it is NP-hard to design a contract providing a multiplicative approximation n1−α—with n being the number of agents—of the principal’s expected utility in an op- timal contract, even when both the number of agents’ actions and the dimensionality of the outcomes are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Next, we complement such a negative result by providing a polynomial-time approximation algorithm for the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In particular, we exploit the reduction to matroid optimization introduced in Section 3 and a result by Sviridenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2017) in order to design an algorithm that, with high probability, outputs a contract providing a multiplicative approx- imation (1−1/e) of the principal’s reward in an optimal contract, up to a small additive loss ǫ > 0, in time polynomial in the instance size and 1/ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Finally, we conclude the paper, in Section 6, by providing our results on Bayesian principal-multi-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' First, we extend the model recently introduced by Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2022b) to our multi-agent setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The key feature of such a model is that, by taking inspiration from classical mechanism design, it adds a type-reporting stage in which each agent is asked to report their type to the principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In such a setting, the principal is better off committing to a menu of randomized contracts rather than a single contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This specifies a collection of probability distributions over (non-randomized) contracts, where each distribution is employed to draw a contract upon a different combination of types reported by the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Surprisingly, we show that it is possible to implement a polynomial-time approximation algorithm for the problem of computing an optimal menu of randomized contracts, whose guarantees (almost) match those obtained for non-Bayesian instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In order to obtain the result, we first provide a characterization of the principal’s optimization problem, by showing that it can be approximately solved by means of a linear program (LP) with polynomially-many variables and exponentially-many constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Notice that this step is non-trivial, since in general the principal’s optimization problem may not admit a maximum, but only a supremum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Our algorithm is based on an ad hoc implementation of the ellipsoid method, which approximately solves such an LP, provided that it has access to a suitably-defined, polynomial-time approximate separation oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Such an oracle can be implemented by using the algorithms developed in Sections 4 and 5 for non-Bayesian instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='2 Related Works Next, we survey the most-related computational works on hidden-action principal-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Works on Principal-Agent Problems with a Single Agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Most of these works focus on non-Bayesian settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Among the most related to ours, Dutting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2021) and D¨utting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2022) study models whose underlying struc- ture is combinatorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In particular, the latter analyze the case in which the outcome space is defined implicitly through a succinct representation, while the former address settings in which the agent selects a subset of actions (rather than a single one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, Babaioff and Winter (2014) study the complexity of contracts in terms of the number of different payments that they specify, while D¨utting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2019) use the computational lens to analyze the efficiency (in terms of principal’s expected utility) of linear contracts with respect to optimal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Recently, some works also considered the more realistic Bayesian settings (Guruganesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Alon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2022a,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In particular, Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2022c) introduce the idea of menus of randomized contracts, showing that in Bayesian settings they enjoy much nicer computational properties than menus of deterministic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', non-randomized) contracts, which were previously studied in (Guruganesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Alon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Works on Principal-Agent Problems with Multiple Agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' All the previous works on multi-agent settings are limited to non-Bayesian instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Babaioff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2006) are the first to study a model with multiple agents (see also its extended version (Babaioff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2012) and its follow-ups (Babaioff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2009, 2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' They study a setting in which agents have binary actions, called effort and no effort, and the outcome is determined according to a proba- bility distribution that depends on the set of agents that decide to undertake effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This model induces externalities among the agents, since the realized outcome (and, in turn, the agents’ payments) depends on the actions taken by all the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Babaioff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2006) show that finding an optimal contract is #P-complete even when the outcome- determining function is represented as a “simple” read-once network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Emek and Feldman (2012) extend the work by Babaioff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2006) by showing that the problem is NP-hard even for a special class of submodular functions, 3 ARXIV PREPRINT - FEBRUARY 1, 2023 while admitting an FPTAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Finally, a very recent preprint by Duetting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2022) considerably extends previous works by providing constant-factor approximation algorithms for problems with submodular and XOS rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2 The Principal–Multi–Agent Problem An instance of the principal-multi-agent problem is characterized by a tuple (N, Ω, A),1 where: N is a finite set of n := |N| agents;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Ω is a finite set of m := |Ω| possible (individual) outcomes of an agent’s action, and A is a finite set of ℓ := |A| actions available to each agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='2 For each agent i ∈ N, we introduce Fi,a ∈ ∆Ω to denote the probability distribution over outcomes Ω induced by action a ∈ A of agent i,3 while ci,a ∈ [0, 1] denotes the agent’s cost for playing such action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='4 For ease of presentation, we let Fi,a,ω be the probability that Fi,a assigns to ω ∈ Ω, so that it holds � ω∈Ω Fi,a,ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We define a ∈ An :=×i∈N A as a tuple of agents’ actions, whose i-th component is denoted by ai and represents the action played by agent i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, we let ω ∈ Ωn :=×i∈N Ω be a tuple of outcomes, whose i-th component ωi is the individual outcome achieved by agent i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Each tuple ω ∈ Ωn has an associated reward to the principal, which we denote by rω ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' As a result, whenever the agents play the actions defined by a tuple a ∈ An, the principal achieves an expected reward equal to Ra := � ω∈Ωn rω � i∈N Fi,ai,ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Notice that, in our model, the principal observes all the elements in the tuple of outcomes ω ∈ Ωn reached by the agents, which consists in an individual outcome ωi for each agent i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This is in contrast with previous works on principal-multi-agent problems (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', (Babaioff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Emek and Feldman, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Duetting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2022)), which assume that the principal can only observe a single outcome that is jointly determined by the tuple of all the agents’ actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='1 Contracts and Principal’s Optimization Problem In a principal-multi-agent problem, the goal of the principal is to maximize their expected utility by committing to a contract, which specifies payments from the principal to each agent contingently on the actual individual outcome achieved by the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Formally, a contract is defined by a matrix p ∈ Rn×m + , whose entries pi,ω ≥ 0 define a payment for each agent i ∈ N and outcome ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='5 Notice that the assumption that payments are non-negative (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', they can only be from the principal to agents) is common in contract theory, where it is known as limited liability (Carroll, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' When agent i ∈ N selects an action a ∈ A under a contract p ∈ Rn×m + , the expected payment from the principal to agent i is Pi,a := � ω∈Ω Fi,a,ω pi,ω, while the agent’s expected utility is Pi,a − ci,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given a contract p ∈ Rn×m + , each agent i ∈ N selects an action such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' it is incentive compatible (IC), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', it maximizes their expected utility among actions in A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' it is individually rational (IR), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', it has non-negative expected utility (if there is no IR action, then agent i abstains from playing so as to maintain the status quo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For ease of presentation, we make the following w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' assumption: Assumption 1 (Null action).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' There exists an action a∅ ∈ A such that ci,a∅ = 0 for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Such an assumption implies that each agent has an action providing them with a non-negative utility, thus ensuring that any IC action is also IR and allowing us to focus w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' on incentive compatibility only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In the following, given a contract p ∈ Rn×m + , we denote by A∗ i (p) ⊆ A the set of actions that are IC for agent i ∈ N under that contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Formally, it holds A∗ i (p) := arg maxa∈A {Pi,a − ci,a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Furthermore, given an action a ∈ A of agent 1For ease of notation, in this paper we assume that all the numerical quantities that define a principal-multi-agent problem instance, such as costs, rewards, and probabilities, are attached to their corresponding elements in the sets N, Ω, and A, so that we can simply write I := (N, Ω, A) to identify an instance of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2For ease of presentation, we assume that all the agents share the same action set and outcome set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Our results continue to hold even if each agent i ∈ N has their own action set Ai and their actions induce outcomes in an agent-specific set Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 3In this paper, given a finite set X, we denote by ∆X the set of all the probability distributions defined over elements of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 4For ease of presentation, costs and rewards are in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' All the results can be easily generalized to an arbitrary range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 5W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', we can restrict the attention to contracts that define payments independently for each agent, rather than dealing with contracts which specify payments based on the tuple of outcomes resulting from the actions of all agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This is because each agent i ∈ N induces a specific outcome ωi with their action (independently of what the others do), and such outcome is observed by the principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' As a consequence, we also have that, in our setting, there are no externalities among the agents, since an agent’s expected utility does not depend on the actions played by other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 4 ARXIV PREPRINT - FEBRUARY 1, 2023 i ∈ N, we let Pi,a ⊆ Rn×m + be the set of contracts such that action a is IC for agent i under them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' formally, Pi,a := � p ∈ Rn×m + | a ∈ A∗ i (p) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given a contract p ∈ Rn×m + , the resulting set A∗ i (p) of IC actions for an agent i ∈ N may contain more than one action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Thus, it is necessary to adopt a suitable tie-breaking-rule assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Remark 1 (On classical tie-breaking rules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Most of the works on principal-agent problems usually assume that, when- ever an agent is indifferent among multiple IC actions, they break ties in favor of the principal (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', (Dutting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Such an assumption is unreasonable in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Indeed, as we show in Corollary 2, the problem of com- puting a utility-maximizing tuple of agents’ actions that are IC under a given contract p ∈ Rn×m + is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We circumvent the issue of classical tie-breaking rules highlighted in Remark 1 by slightly abusing terminology and extending the notion of contract to also include action recommendations for the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Formally, we identify a contract with a pair (p, a∗), where p ∈ Rn×m + defines the payments and a∗ ∈×i∈N A∗ i (p) specifies a tuple of agents’ actions, which should be interpreted as action recommendations suggested by the principal to the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given that the actions in a∗ are IC under p, we assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' that the agents stick to such recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In conclusion, the principal’s optimization problem reads as follows: Definition 1 (Principal’s Optimization Problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given an instance (N, Ω, A) of principal-multi-agent problem, com- pute an optimal contract (p, a∗)—with p ∈ Rn×m and a∗ ∈ ×i∈N A∗ i (p)—, which is defined as a pair (p, a∗) maximizing the principal’s expected utility: Ra∗ − � i∈N Pi,a∗ i = � ω∈Ωn rω � i∈N Fi,ai,ωi − � i∈N � ω∈Ω Fi,a,ω pi,ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='2 On the Representation of Principal’s Rewards Representing principal’s rewards in a principal-multi-agent problem becomes unfeasible when there are many agents, since the number of possible tuples of outcomes grows as mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Thus, we work with a succinct representation of principal’s rewards, which we formally introduce in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We remark that, with arbitrary rewards, an optimal contract can be found in time polynomial in the instance size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', in time depending polynomially on mn), as we show in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We say that a principal-multi-agent problem instance (N, A, Ω) has succinct rewards if: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' outcomes can be represented as non-negative q-dimensional vectors with q ∈ N>0 representing the dimen- sionality of the outcome space, namely Ω is a finite subset of Rq +;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' the principal’s rewards can be expressed by means of a reward function g : Rnq + → R such that rω = g(ω) holds for every tuple of outcomes ω ∈ Ωn, and, thus, we can also write Ra = � ω∈Ωn g(ω) � i∈N Fi,ai,ωi for every a ∈ An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='6 Let us remark that, for ease of presentation and overloading notation, we denote tuples of outcomes as vectors, namely ω ∈ Ωn ⊆ Rnq + , where we let ωi,j be the j-th component of the vector that identifies the outcome achieved by agent i, for all i ∈ N and j ∈ [q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='7 Moreover, in the following, we assume that g : Rnq + → R can be accessed through an oracle that, given ω ∈ Ωn, outputs g(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='8 In this work, we make the following common assumption on principal’s rewards: Assumption 2 (Increasing rewards).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The principal’s reward function g : Rnq + → R is increasing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' formally, it holds that g(ω) ≥ g(ω′) for all ω, ω′ ∈ Rnq + : ω ≥ ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, we will focus on two particular classes of reward functions, which, as we show next, enjoy some useful properties and are met in many real-world settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 6Notice that, since we assume that rω ∈ [0, 1] for all ω ∈ Ωn, while the function g is allowed to take any real value over its domain Rnq + , it has to hold that g(ω) ∈ [0, 1] for all ω ∈ Ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 7In this paper, given a positive natural number x ∈ N>0, we let [x] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' , x} be the set of the first x natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 8In this paper, for ease of exposition, we assume that the value of Ra for any given a ∈ An can be computed in polynomial time, without enumerating tuples of outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The value of Ra can be approximated up to any arbitrarily small error with high probability by sampling each ωi independently from Fi,a, and evaluating g(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' All the results in the paper can be easily extended to also account for this additional (arbitrarily small) approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 5 ARXIV PREPRINT - FEBRUARY 1, 2023 Definition 2 (DR-submodularity and IR-supermodularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' A reward function g : Rnq + → R is diminishing-return submodular (DR-submodular) if, for all ω, ω′, ω′′ ∈ Rnq + : ω ≤ ω′, it holds g(ω + ω′′) − g(ω) ≥ g(ω′ + ω′′) − g(ω′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, a reward function g : Rnq + → R is increasing-return supermodular (IR-supermodular)if its opposite function −g is DR-submodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Let us remark that, when the reward function g is continuously differentiable, then the property that characterizes DR-submodular functions has a more intuitive interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Indeed, as shown by Bian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2017), when g is continuously differentiable, g is DR-submodular if and only if: ∇g(ω) ≥ ∇g(ω′) ∀ω, ω′ ∈ Rnq + : ω ≥ ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Intuitively, this means that, if a tuple of outcomes ω′ dominates component-wise another tuple ω, then in ω′ the reward function grows slower than in ω along all of its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This property is satisfied in many real-world scenarios, as we show in the following specific example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Example 1 (Selling multiple products).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Consider a principal-agent problem modeling the interaction between a firm and a salesperson (the example can be easily generalized to the case of multiple salespersons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The firm wants to sell q ∈ N>0 different products, and the salesperson can sell a variable quantity of each product, depending on the level of effort put in selling each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Thus, the outcome achieved by the salesperson can be encoded by a vector ω ∈ R1q + whose j-th component ω1,j represents the quantity of product j being sold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In such a setting, a DR-submodular reward function g models scenarios in which the firm is subject to diseconomies of scale, and, thus, the marginal return of each unit of product sold decreases as the quantity sold increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This may be due to the fact that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', the firm has to sustain much higher operational costs in order increase its selling capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' On the other hand, an IR-supermodular reward function g models cases in which there are economies of scale, and, thus, the marginal return of each unit of product sold increases with quantity (since, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', the fixed costs are more efficiently covered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 3 Reducing Principal–Multi–Agent Problems to Matroids In this section, we show that computing an optimal contract in principal-multi-agent problems can be reduced in polynomial time to a maximization problem defined over a special class of matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' First, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='1, we introduce some preliminary definitions on matroids and optimization problems over matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='2, we provide the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We conclude the section with Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='3, in which we provide a preliminary technical result for the problem of max- imizing functions defined over 1-partition matroids and satisfying a particular (stronger) notion of supermodularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This result will be useful in the following Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='1 Preliminaries on Matroids A matroid M := (G, I) is defined by a finite ground set G and a collection I of independent sets, which are subsets of G satisfying some characteristic properties, namely: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' the empty set is independent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', ∅ ∈ I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' every subset of an independent set is independent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', for S′ ⊆ S ⊆ G, if S ∈ I then S′ ∈ I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' if S ∈ I and S′ ∈ I are two independent sets such that S has more elements than S′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', |S| > |S′|, then there exists an element x ∈ S \\ S′ such that S′ ∪ {x} ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Any subset S ⊆ G such that S /∈ I is said to be dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The bases of the matroid M are all the maximal independent sets of M, where an independent set is said to be maximal if it becomes dependent by adding any element of G to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We denote by B(M) ⊂ 2G the set of the bases of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We refer the reader to (Schrijver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2003) for a detailed treatment of matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In the following, we will also consider optimization problems defined over matroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In particular, given a set function f : 2G → R assigning a value to each subset of the ground set, the associated maximization problem over a matroid M := (G, I) is defined as maxS∈I f(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='2 Reduction to Matroid Optimization In order to provide our reduction, we need to introduce the following class of matroids: 6 ARXIV PREPRINT - FEBRUARY 1, 2023 Definition 3 (1-Partition Matroid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' A matroid M := (G, I) is a 1-partition matroid if there exists d ∈ N+ subsets Gi ⊆ G of ground elements such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' G = � i∈[d] Gi and Gi ∩ Gj = ∅ for all i, j ∈ [d] : i ̸= j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' I = {S ⊆ G : |S ∩ Gi| ≤ 1 ∀i ∈ [d]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Intuitively, in a 1-partition matroid, the ground set G is partitioned into d disjoint subsets Gi, and the independent sets are all and only the subsets of G that contain at most one element of each subset Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In the following, we denote by M := ({Gi}i∈[d] , I) a 1-partition matroid with G := � i∈[d] Gi, and, for ease of notation, we let ki := |Gi| for every i ∈ [d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Notice that, as it is immediate to check, the set B(M) of the bases of a 1-partition matroid M := ({Gi}i∈[d] , I) is made by all the subsets of G containing exactly one element for each subset Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Next, we show how the problem of computing an optimal contract in principal-multi-agent problems can be reduced in polynomial time to a maximization problem defined over a suitably-constructed 1-partition matroid, which is formally defined as follows: Definition 4 (Mapping from principal-multi-agent problems to 1-partition matroids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given an instance of principal- multi-agent problem, say I := (N, Ω, A), we define its corresponding 1-partition matroid MI := ( � GI i � i∈N , II) as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' GI i := {(i, a) : a ∈ A} for all i ∈ N, with GI := � i∈[d] GI i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' II := � S ⊆ GI : |S ∩ GI i | ≤ 1 ∀i ∈ [d] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' It is immediate to check that MI is indeed a 1-partition matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, its bases correspond one-to-one to agents’ action profiles a ∈ An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In particular, an independent set S ∈ II of MI assigns an action to each agent in NS := � i ∈ N : |S ∩ GI i | = 1 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' we denote by aS,i ∈ A the action associated to agent i ∈ NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Since a base of a 1-partition matroid is any independent set S ∈ II containing one element for each Gi, it completely specifies an agents’ action profile, which we denote by aS = (aS,i)i∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For ease of presentation, in the following we overload notation and write aS = (aS,i)i∈N also for independent sets S ∈ II that are not bases, by letting all the unspecified actions be equal to the null one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' formally, aS,i = a∅ for all i ∈ N \\ NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The following theorem formalizes our reduction: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given an instance I := (N, Ω, A) of principal-multi-agent problem, the problem of computing a contract maximizing the principal’s expected utility can be reduced in polynomial time to solving maxS∈II f I(S) over the 1-partition matroid MI = ( � GI i � i∈N , II), where f I : 2GI → R is a set function such that, for every independent set S ∈ II, it holds: f I(S) := RaS − � i∈N �Pi,aS,i, where �Pi,aS,i = min p∈Pi,aS,i � ω∈Ω Fi,aS,i,ω pi,ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Intuitively, f I(S) is equal to the maximum possible expected utility that the principal can get by means of contracts under which the actions in aS are IC and are those recommended by the principal to the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The proof of Theorem 1 relies on the following useful lemma, which shows that the optimal value of f I is always attained at a base of MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' There always exists a base S∗ ∈ B(MI) of MI such that f I(S∗) = maxS∈II f I(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Let us also remark that Lemma 1 and Theorem 1 immediately provide a polynomial-time algorithm for finding an optimal contract in principal-multi-agent instances without succinct rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Indeed, since the optimal value of f I is always attained at least one base of the matroid MI (Lemma 1), in order to find an optimal contract it is sufficient to enumerate all the bases of MI, which are mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Without a succinct reward representation, the size of an instance of principal-multi-agent problem grows as mn (there is a reward value for each tuple of agents’ outcomes), and, thus, the enumerative algorithm runs in time polynomial in the instance size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='3 Preliminary Technical Results on 1-Partition Matroids We first introduce a particular class of set functions defined over 1-partition matroids, which we call ordered- supermodular functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In order to do this, we first need some additional notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given a 1-partition matroid M := ({Gi}i∈[d] , I), for each i ∈ [d] we introduce a bijective function πi : [ki] → Gi to denote an ordering of the subset Gi in which the elements are ordered from πi(1) to πi(ki).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given two independent sets S, S′ ∈ I of the matroid, 7 ARXIV PREPRINT - FEBRUARY 1, 2023 we denote by S ∧ S′ the partition-wise “maximum” of the two sets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', the set made by an element x ∈ (S ∪ S′) ∩ Gi with maximal value of π−1 i (x) for each partition i ∈ [d] (notice that (S ∪ S′) ∩ Gi contains at most one element of Gi for each of the two sets S and S′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Analogously, we define S ∨ S′ as the partition-wise “minimum” of the two sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, a set function is said ordered-supermodular if there exist some orderings of the sets Gi such that the function satisfies the classical condition of supermodularity over the independent sets of the matroid, with the usual union and intersection operators replaced by the partition-wise “maximum” ∧ and “minimum” ∨, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Formally: Definition 5 (Ordered-supermodular function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' A set function f : 2G → R defined over a 1-partition matroid M := ({Gi}i∈[d] , I) is said to be ordered-supermodular if there exist bijective functions πi : [ki] → Gi for i ∈ [d] such that, for every pair of independent sets S′, S ∈ I: f(S ∧ S′) + f(S ∨ S′) ≥ f(S) + f(S′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Notice that, if one restricts the attention to independent sets S′, S ∈ I such that S ∪ S′ ∈ I, then the condition for ordered-supermodularity coincides with that for supermodularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Thus, intuitively, the former can be seen as a way of tightening the latter in order to also account for cases in which the union of independent sets is not independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Finally, we show that the characteristic feature of ordered-supermodular functions allows us to reduce their optimiza- tion to solving maximization problems of supermodular functions defined over rings of sets, which can be done in polynomial time (Schrijver, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Bach, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='9 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The problem of maximizing an ordered-supermodular function over a 1-partition matroid can be reduced in polynomial time to maximizing a supermodular function over a ring of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The problem of maximizing an ordered-supermodular function over a 1-partition matroid admits a polynomial-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 4 Principal-Multi-Agent Problems with IR-supermodular Rewards In this section, we study principal-multi-agentproblems with succinct rewards specified by IR-supermodularfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' First, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='1, we prove that in such setting the problem of computing an optimal contract is inapproximable in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='2 we show that, under mild assumptions, the problem can be solved in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='1 Inapproximability Result In order to prove the negative result, we provide a reduction from the LABEL-COVER problem, which consists in assigning labels to the vertexes of a bipartite graph in order to satisfy some given constraints that define which pairs of labels can be assigned to vertexes connected by an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In particular, we consider the promise version of the problem, in which, given an instance such that either there exists an assignment of labels satisfying at least a fraction c of the constraints or all the possible assignments satisfy less than a fraction s of them (with s ≤ c), one has to establish which one of the two cases indeed holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Such a problem is known to be NP-hard (Raz, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Arora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We refer the reader to Appendix B for a formal definition of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Our inapproximability result formally reads as follows: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For any constant ρ > 0, in principal-multi-agent problems with succinct rewards specified by an IR- supermodular function, it is NP-hard to design a contract providing a ρ-approximation of the principal’s expected utility in an optimal contract, even when both the number of outcomes m and the number of agents’ actions ℓ are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Indeed, the proof of Theorem 3 provides an even stronger hardness result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' It also shows that it is NP-hard to find a tuple of agents’ actions a ∈ An that is “approximately” optimal for the principal under a given contract p ∈ Rn×m + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Formally, the following corollary holds: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For any constant ρ > 0, in principal-multi-agent problems with succinct rewards specified by an IR- supermodular function, it is NP-hard to compute a ρ-approximate solution to the problem of finding the best (for the principal) tuple of IC agents’ actions a ∈×i∈N A∗ i (p) for a given contract p ∈ Rn×m + , even when both the number of outcomes m and that of agents’ actions ℓ are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Corollary 2 is readily proved by noticing that the proof of Theorem 3 continues to hold even if we restrict it to the null contract in which all the payments are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 9We recall that a ring of sets is a family of sets R that is closed under both union and intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Formally, given any two sets S, S′ ∈ R, it holds S ∪ S′ ∈ R and S ∩ S′ ∈ R (Birkhoff, 1937).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 8 ARXIV PREPRINT - FEBRUARY 1, 2023 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='2 A Polynomial-time Algorithm for Instances Satisfying the FOSD Condition In the following, we show how to circumvent the negative result established by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In particular, we prove that, in principal-multi-agent problems with succinct rewards specified by an IR-supermodular function, under some mild additional assumptions the problem of computing an optimal contract can indeed be solved in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We consider instances satisfying a particular first-order stochastic dominance (FOSD) condition, which is similar to several properties that are commonly studied in the contract theory literature (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', (Tadelis and Segal, 2005)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, such a condition is reasonably satisfied in many real-world settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Intuitively, it states that the higher the cost of an agent’s action, the bigger the probability with which such an action induces “good” outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For instance, in salesperson problems with multiple products (Example 1), such a condition is always satisfied, since outcome vectors represent the quantity of each product being sold and action costs encode the effort levels undertaken by the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Naturally, the salesperson undertaking an higher level of effort in selling products will result in a bigger probability of generating large volumes of sales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In order to formally define the FOSD condition, we first need to introduce some additional notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given a subset of outcomes Ω′ ⊆ Ω, we say that Ω′ is comprehensive whenever, for every ω ∈ Ω′ and ω′ ∈ Ω, if ω′ ≤ ω then ω′ ∈ Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, for ease of presentation, with a slight abuse of notation and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' we assume that the actions of each agent i ∈ N are re-labeled so that A = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' , aℓ} with ci,aj ≤ ci,aj+1 for every j ∈ [ℓ − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, we have the following definition: Definition 6 (First order stochastic dominance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' An instance of principal-multi-agent problem is said to satisfy the first-order stochastic dominance (FOSD) condition if, for every agent i ∈ N and action index j ∈ [ℓ−1], the following holds for all the comprehensive sets Ω′ ⊆ Ω: � ω∈Ω′ Fi,aj+1,ω ≤ � ω∈Ω′ Fi,aj,ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' A condition similar to Definition 6, called monotone likelihood ratio property (MLRP), has been consid- ered by D¨utting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2019) limited to the case in which outcomes are identified by scalar values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In such settings, the MLRP is strictly stronger than the FOSD condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Our definition of FOSD generalizes the classical FOSD condition (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', (Tadelis and Segal, 2005)) from settings in which the outcomes are scalar values to those where they are vector values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Next, we prove how to design a polynomial-time algorithm for the problem of finding an optimal contract by exploiting the FOSD condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Intuitively, the idea of the proof is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' First, thanks to Theorem 1, we can reduce in polynomial time an instance I := (N, A, Ω) of the principal-multi-agent problem to the optimization of a suitably- defined set function f I over a 1-partition matroid MI (see Theorem 1 and Definition 4 for the definition of f I and MI, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, by Corollary 1, if f I is ordered-supermodular we can solve the optimization problem in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Hence, in order to prove the result, we simply need to show that, whenever the FOSD condition is satisfied, the function f I is indeed ordered-supermodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' First, we prove the following preliminary result which follows from (Østerdal, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In principal-multi-agent problems with succinct rewards that satisfy the FOSD condition, for every agent i ∈ N and pair aj, ak ∈ A of agent i’s actions such that j < k, there exists a collection of probability distributions µω ∈ ∆Ω−, one per outcome ω ∈ Ω, which are supported on the finite subset of the positive orthant Ω− := Rm + ∩ {ω − ω′ | ω, ω′ ∈ Ω} and satisfy the following equations: Fi,ak,ω = � ω′∈Ω Fi,aj,ω′ µω′ ω−ω′ ∀ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given Lemma 2, we are ready to show that, if the instance I meets the FOSD condition, then its corresponding set function f I is indeed ordered-supermodular over the 1-partition matroid MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given an instance I := (N, Ω, A) of principal-multi-agent problem that (i) has succinct rewards specified by an IR-supermodular function and (ii) satisfies the FOSD condition, the set function f I defined over the 1-partition matroid MI = ( � GI i � i∈N , II) is ordered-supermodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Finally, Lemma 3 allows us to prove the main positive result of this section: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For principal-multi-agent problem instances that (i) have succinct rewards specified by an IR- supermodular function and (ii) satisfy the FOSD condition, the problem of computing an optimal contract admits a polynomial-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 9 ARXIV PREPRINT - FEBRUARY 1, 2023 5 Principal-Multi-Agent Problems with DR-submodular Rewards In this section, we switch the attention to principal-multi-agent problems with succinct rewards specified by DR- submodular functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' First, similarly to the case of IR-supermodular reward functions, we provide a strong negative result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In particular, we show that the problem of computing an optimal contract cannot be approximated up to within any constant factor, even when either the number of actions or the dimensionality of outcome vectors is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In order to prove the negative result, we provide a reduction from the promise version of the well-known INDEPENDENT-SET problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In such a version of the problem, one is given an undirected graph G := (V, E) such that either there exists an independent set of size at least |V |1−α—for some α > 0—or all the independent sets have size at most |V |α, and is asked to decide which one of the two cases holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Such a problem is known to be NP- hard for any α > 0 (H˚astad, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Zuckerman, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This is exploited by our reduction in order to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The reader can find more details on the definition of the promise version of INDEPENDENT-SET in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For any constant α > 0, in principal-multi-agent problems with succinct rewards specified by a DR- submodular function, it is NP-hard to design a contract providing an n1−α–approximation of the principal’s expected utility in an optimal contract, even when both the number of agents’ actions ℓ and the dimensionality q of outcome vectors are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Next, we complement the inapproximability result in Theorem 5 by providing a polynomial-time approximation al- gorithm for the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In order to do so, we exploit the fact that, in settings with succinct rewards specified by DR-submodular functions, the set function f I constructed in Theorem 1 is always a submodular function over the 1-partition matroid MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' However, this is not sufficient, since such a function is non-monotone and non-positive, and, thus, we need to deploy some non-standard tools in order to come up with a polynomial-time approximation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' As a first step, given an instance I := (N, Ω, A) of principal-multi-agent problem, we extend the definition of the function f I to all the subsets of GI (notice that Theorem 1 provides a value of f I only for the independent sets I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' To do this, we first need to introduce some additional notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For ease of presentation, in the rest of this section we will make the following w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' assumption: Assumption 3 (Null outcome).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' There exists an outcome ω∅ ∈ Ω such that ω∅ = 0 ∈ Rq and, for every agent i ∈ N, it holds that Fi,a∅,ω∅ = 1 and Fi,a,ω∅ = 0 for all a ∈ A \\ {a∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, by slightly abusing notation, given any S ⊆ GI we let Fi,S := � (i,a)∈S∪{(i,a∅)|i∈N} Fi,a be the probability distribution of the sum of independent random variables distributed as Fi,a, one for each pair (i, a) in S ∪ {(i, a∅) | i ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Notice that the probability distributions defined above are no longer supported on the set of outcomes Ω, but rather on the set of all the possible vectors in Rq + that can be obtained as the sum of at most nℓ (possibly repeated) vectors in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We denote such a set by ˜Ω ⊆ Rq +, and let Fi,S,ω be the probability that Fi,S assigns to ω ∈ ˜Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, we let ˜Ωn :=×i∈N ˜Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Finally, we overload notation and let RS := � ω∈˜Ωn rω � i∈N Fi,S,ωi for any S ⊆ GI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Notice that, since any independent set S ∈ II includes at most one pair (i, a) for each agent i ∈ N, it is easy to check that RS = RaS (see Section 4 for the definition of RaS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We are now ready to provide the formal definition of the extension of f I: Definition 7 (Extension of f I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given an instance I := (N, Ω, A) of principal-multi-agent problem, the extension of f I to all the subsets of GI is such that, for every S ⊆ GI: f I(S) := RS − � (i,a)∈S �Pi,a, where �Pi,a := min p∈Pi,a � ω∈Ω Fi,a,ω pi,ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The crucial result that we need in order to design a polynomial-time approximation algorithm is the following Lemma 4, which shows that the extended function f I can be decomposed as the sum of a monotone-increasing submodular function and a linear one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Formally: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given an instance I := (N, Ω, A) of principal-multi-agent problem with succinct rewards specified by a DR-submodular function, the extended set function f I (see Definition 7) can be defined as f I(S) := fI(S) + lI(S) for every S ⊆ GI, where fI : 2GI → R+ is a monotone-increasing submodular function and lI : 2GI → R is a linear function, both defined over the 1-partition matroid MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Lemma 4 allows us to apply a result by Sviridenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2017), who provide a polynomial-time approximation algo- rithm for the problem of optimizing the sum of a monotone-increasing submodular function and a linear one over a matroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This immediately gives the following result: 10 ARXIV PREPRINT - FEBRUARY 1, 2023 Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In principal-multi-agent problems with succinct rewards specified by a DR-submodular function, the problem of computing an optimal contract admits a polynomial-time approximation algorithm that, for any ǫ > 0 given as input, outputs a contract with principal’s expected utility at least (1 − 1/e)R(p,a∗) − P(p,a∗) − ǫ for any contract (p, a∗) with high probability, where R(p,a∗) ∈ [0, 1], respectively P(p,a∗) ∈ R+, denotes the expected reward, respectively payment, under (p, a∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 6 Bayesian Principal-multi-agent Problems In this last section, we study Bayesian principal-multi-agent problems in which each agent has a private type determin- ing their action costs and distributions over outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In particular, we extend the Bayesian model recently introduced by Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2022b) to multi-agent settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' First, in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='1 we formally introduce Bayesian principal-multi-agent problems and all their related concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='2 provides a formulation of the computational problem that the principal has to solve in Bayesian settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Next, in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='3 we show how such a problem can be “approximately formulated” as an LP with exponentially-many variables and polynomially-many constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Finally, in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='4 we exploit such a formu- lation to design a polynomial-time approximation algorithm for the problem, based on an ad hoc implementation of the ellipsoid method that uses an approximate separation oracle that can be implemented in polynomial time in settings having the same properties as those in which we derived our positive results in Sections 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='1 The Model An instance of the Bayesian principal-multi-agent problem is characterized by a tuple (N, Θ, Ω, A), where N, Ω, and A are defined as in non-Bayesian instances, while Θ is a finite set of agents’ types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='10 We denote by θ ∈ Θn :=×i∈N Θ a tuple of agents’ types, whose i-th component θi represents the type of agent i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We assume that agents’ types are jointly determined according to a probability distribution λ ∈ ∆Θn supported on a subset supp(λ) ⊆ Θn of tuples of agents’ types—with λθ being the probability assigned to θ ∈ supp(λ)—, and that such a distribution is commonly known to the principal and all the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='11 Action costs and distributions over outcomes are extended so that they also depend on the agent’s type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' formally, they are denoted as Fi,θ,a and ci,θ,a, where θ ∈ Θ is the type of agent i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Similarly, we extend the definition of expected reward, denoted as Rθ,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', we modify Assumption 1 so that the null action a∅ now satisfies ci,θ,a∅ = 0 for all i ∈ N and θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Finally, for an agent i ∈ N of type θ ∈ Θ, we define A∗ i,θ(p) ⊆ A as the set of actions that are IC under a given contract p ∈ Rn×m + , while Pi,θ,a ⊆ Rn×m + denotes the set of contracts under which a given action a ∈ A is IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Following the line of Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2022b), we consider the case in which the principal commits to a menu of randomized contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In our multi-agent setting, a randomized contract is defined as a probability distribution γ supported on Rn×m + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, a menu consists in a collection Γ = (γθ)θ∈Θn containing a randomized contract γθ for each possible tuple of agents’ types θ ∈ Θn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The interaction between the principal and agents having types specified by θ ∼ λ goes as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' the principal commits to a menu of randomized contracts Γ = (γθ)θ∈Θn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' each agent i ∈ N reports a type ˆθi ∈ Θ to the principal (possibly different from their type θi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' the principal draws a contract p ∼ γ�θ, where �θ ∈ Θn denotes the tuple of agents’ types whose i-th component is the type ˆθi reported by agent i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' each agent i ∈ N plays an IC action ai ∈ A∗ i,θi(p) according to their true type θi, resulting in a tuple of agents’ actions a ∈×i∈N A∗ i,θi(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' As discussed in Section 2 (see Remark 1), in our multi-agent setting a contract does not only need to specify payments, but also action recommendations for the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Thus, in the rest of this section, whenever we refer to a contract 10For ease of exposition, all agents share the same set Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Our results can be easily extended to the case of agent-specific sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 11Let us remark that, as it is the case for action costs and distributions over outcomes, as well as rewards, the probabilities defining the distribution λ are part of the representation of a Bayesian principal-multi-agent problem instance, and, thus, they are part of the input to the principal’s optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Hence, the running time of any polynomial-time algorithm for such a problem must depend polynomially on the size of supp(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' It is crucial that only probabilities λθ corresponding to tuples of agents’ types θ ∈ supp(λ) in the support of λ are specified as input, otherwise the size of the input representation would always be exponential in n, rendering the task of designing polynomial-time algorithms straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 11 ARXIV PREPRINT - FEBRUARY 1, 2023 p ∈ Rn×m + belonging to the support of a randomized contract γθ for θ ∈ Θn, we always assume that it is paired with a tuple a∗ ∈×i∈N A∗ i,θi(p) of IC (for the types specified by θ) action recommendations for the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In a Bayesian setting, the goal of the principal is to commit to an optimal menu of randomized contracts, which is one maximizing their expected utility, which is obtained by extending the non-Bayesian expression in Definition 1 to also account for the expectation with respect to the distribution λ of agents’ types and the distributions γθ defining the randomized contracts in the menu (see Objective (1a) below for a formal mathematical formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' As in single-agent settings (Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2022b), it is possible to focus w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' on menus of randomized contracts that are dominant-strategy incentive compatible (DSIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='12 These are menus such that the agents are always incen- tivized to truthfully report their type to the principal, no matter the types reported by others (see Constraints (1b) for a formalization of the DSIC conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='2 Formulating the Principal’s Optimization Problem Next, we show how to formulate the problem of computing an optimal DSIC menu of randomized contacts in Bayesian principal-multi-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The formulation that we propose in the following is specifically tailored so as to ease the design of our approximation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' As a first step, we show that we can focus w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' on randomized contracts γθ having a finite support supp(γθ) ⊆ Rn×m + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Such a result is already known for single-agent settings (see Lemma 1 in (Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2022b) and Theorem 1 in (Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2022)), but it can be easily generalized to our multi-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In particular, since in our model there are no externalities among the agents, it is immediate to adapt the results of Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2022b) and Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2022) in order to show that there always exists an optimal DSIC menu of randomized contracts such that, for every agent i ∈ N and tuple of agents’ types θ ∈ Θn, the contracts in the support supp(γθ) of γθ specify at most one different agent i’s payment scheme for each action a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, such agent i’s payment scheme is such that action a is IC when the type of agent i is θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Formally: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given an instance I := (N, Θ, Ω, A) of Bayesian principal-multi-agent problem and a DSIC menu of randomized contracts, there always exists another DSIC menu of randomized contracts Γ = (γθ)θ∈Θn with at least the same principal’s expected utility such that, for every i ∈ N and θ ∈ Θn, it holds that ��� pi | p ∈ supp(γθ) ∧ p ∈ Pi,θi,a��� ≤ 1 for all a ∈ A, where pi ∈ Rm + denotes the i-th row of matrix p (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', the agent i’s payment scheme under contract p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Lemma 5 allows us to identify the contracts in the support supp(γθ) of γθ with their corresponding tuples of action recommendations for the agents, since there could be at most one different contract for each one of such tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Thus, in order to characterize the elements defining a menu of randomized contracts which are needed for our purposes, it is sufficient to specify: for every tuple of agents’ types θ ∈ Θn and tuple of agents’ actions a ∈ An, the probability tθ,a ∈ [0, 1] that the randomized contract γθ places on the contract whose corresponding action recommendations for the agents are specified by a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' for every agent i ∈ N, tuple of agents’ types θ ∈ Θn, and action a ∈ A, the probability ξi,θ,a ∈ [0, 1] with which agent i is recommended to play action a after the agents collectively reported the types specified by θ to the principal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' for every agent i ∈ N, tuple of agents’ types θ ∈ Θn, action a ∈ A, and outcome ω ∈ Ω, the payment pi,θ,a,ω ≥ 0 from the principal to agent i when the agents reported the types in θ to the principal, agent i is recommended action a, and the realized outcome is ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We are now ready to provide our formulation of the problem of computing an optimal DSIC menu of randomized contracts in Bayesian principal-multi-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Before doing that, for ease of presentation, we introduce some additional notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In particular, we let ˜Θn := supp(λ) be the set of tuples of agents’ types that could be possibly reported to the principal if the agents truthfully reveal their types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, given any θ ∈ Θn, we let θ−i be the tuple obtained by dropping agent i’s type θi from θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, given a type θ ∈ Θ, we write (θ, θ−i) to denote the tuple obtained by adding θ to θ−i as agent i’s type, so that θ = (θi, θ−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Finally, for every agent i ∈ N, we denote by 12It is easy to show that focusing on DSIC menus of randomized contracts is w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' by using a revelation-principle-style argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' See the book by Shoham and Leyton-Brown (2008) for some examples of these kinds of arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 12 ARXIV PREPRINT - FEBRUARY 1, 2023 ˜Θ−i := � θ−i : θ ∈ ˜Θn� the set of all tuples of types that could be possibly reported to the principal by agents other than i, assuming that they truthfully reveal their types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='13 We can now formulate the principal’s optimization problem as follows: sup � θ∈ ˜Θn λθ � a∈An tθ,a Rθ,a − � i∈N � θ∈ ˜Θn λθ � a∈A ξi,θ,a � ω∈Ω Fi,θi,a,ω pi,θ,a,ω s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (1a) � a∈A ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a �� ω∈Ω Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω − ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a � ≥ � a∈A ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a max a′∈A �� ω∈Ω Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω − ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a′ � ∀i ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ ˜Θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ Θ (1b) � a∈A ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a = 1 ∀i ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ−i ∈ ˜Θn −i (1c) � a∈An:ai=a tθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a = ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a ∀i ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ ˜Θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀a ∈ A (1d) tθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a ≥ 0 ∀θ ∈ ˜Θn −i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀a ∈ An (1e) ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a ≥ 0 ∀i ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ−i ∈ ˜Θn −i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀a ∈ A (1f) pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω ≥ 0 ∀i ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ−i ∈ ˜Θn −i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀a ∈ A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀ω ∈ Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (1g) where Objective (1a) is the principal’s expected utility for the menu of randomized contracts encoded by the variables in the problem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Constraints (1b) specify the conditions ensuring that the menu is DSIC (for θ ̸= θi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' as well as the conditions guaranteeing that any action a such that ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a > 0 is IC for an agent i of type θi under the payments defined by variables pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' while Constraints (1c) and (1d) ensure that the menu of randomized contracts is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Notice that Problem (1) is defined in terms of sup rather than max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This is because, as shown in (Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2022b), even in single-agent settings the problem of computing an optimal DSIC menu of randomized contracts may not admit a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In the following, for ease of presentation, we let SUP be the optimal value of Problem (1) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', the value of the supremum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='3 An “Approximately-optimal” LP Formulation As a preliminary step towards the design of our approximation algorithm (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='4), we show how to find an “approximately-optimal” DSIC menu of randomized contracts by solving an LP which features exponentially-many variables and polynomially-many constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In the following, we will make extensive use of the set Ai,θ ⊆ A of actions which are inducible for an agent i ∈ N of type θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This is the set of all actions that are IC for an agent i of type θ under at least one contract;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' formally, Ai,θ := � a ∈ A | ∃p ∈ Rn×m + : a ∈ A∗ i,θ(p) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' First, we can prove the following useful result: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' There exists a function τ : N → R such that τ(x) is O(2poly(x))—with poly(x) being a polynomial in x—and, for every instance I := (N, Θ, Ω, A) of Bayesian principal-multi-agent problem, agent i ∈ N, type θ ∈ Θ, and inducible action a ∈ Ai,θ, there exists a contract p ∈ Rn×m + such that a ∈ A∗ i,θ(p) and pi,ω ≤ τ(|I|) for all ω ∈ Ω, where |I| is the size of instance I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='14 Intuitively, Lemma 6 states that, if an action a is inducible for an agent i of type θ, then there exists a contract under which such an action is IC and whose payments are “small”, in the sense that they can be represented with a number of bits that is upper bounded by a quantity depending polynomially on the size of the problem instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' As we show next, such a result is crucial for proving Theorem 7, as it allows to satisfactorily bound the principal’s expected utility loss due to solving an LP rather than Problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 13Using the sets ˜Θn and ˜Θn −i to index the variables appearing in Problem (1) is crucial in order to guarantee that the number of variables and that of constraints defining the problem is polynomial in the size of supp(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Indeed, indexing the variables over all the tuples of agents’ types in Θn would lead to a number of variables and constraints exponential in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 14In the rest of the section, we always assume that the size of a problem instance is expressed in terms of number of bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 13 ARXIV PREPRINT - FEBRUARY 1, 2023 Next, we formally introduce LP (2), which is obtained from Problem (1) by (i) replacing each product of two variables ξi,θ,a pi,θ,a,ω with a single variable yi,θ,a,ω, (ii) considering the inducible actions in Ai,θ as the only actions available to an agent i of type θ, and (iii) linearizing the max operator in Constraints (1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' By letting An,θ :=×i∈N Ai,θi for every θ ∈ ˜Θn, we can write: max � θ∈ ˜Θn λθ � a∈An,θ tθ,a Rθ,a − � i∈N � θ∈ ˜Θn λθ � a∈Ai,θi � ω∈Ω Fi,θi,a,ω yi,θ,a,ω s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2a) � a∈Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θi �� ω∈Ω yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω − ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a � ≥ � a∈Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ γi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a ∀i ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ ˜Θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ Θ (2b) γi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a ≥ � ω∈Ω yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω − ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a′ ∀i ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ ˜Θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀a ∈ Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀a′ ∈ Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θi (2c) � a∈Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a = 1 ∀i ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ−i ∈ ˜Θn −i (2d) � a∈An,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ:ai=a tθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a = ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a ∀i ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ ˜Θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀a ∈ Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θi (2e) tθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a ≥ 0 ∀θ ∈ ˜Θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀a ∈ An,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ (2f) ξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a ≥ 0 ∀i ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ−i ∈ ˜Θn −i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀a ∈ Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ (2g) yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω ≥ 0 ∀i ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ−i ∈ ˜Θn −i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀a ∈ Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀ω ∈ Ω (2h) γi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a ∀i ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ ˜Θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀a ∈ Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2i) By letting LP be the optimal value of LP (2), the following lemma shows that such a value is always at least as large as the value of the supremum defined in Problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For every instance of Bayesian principal-multi-agent problem, it holds LP ≥ SUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Lemma 7 is proved by showing that, given any feasible solution to Problem (1), it is possible to recover a feasible solution to LP (2) having the same objective function value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' However, the converse is not true in general, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', given a feasible solution to LP (2), it is not always possible to build a feasible solution to Problem (1) having at least the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Thus, it might be the case that SUP < LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This is caused by the existence of what we call irregular feasible solutions to LP (2): Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' A feasible solution to LP (2) is said to be irregular if there exists an agent i ∈ N, a tuple of agents’ types θ ∈ Θ, an inducible action a ∈ Ai,θi, and an outcome ω ∈ Ω such that yi,θ,a,ω > 0 and ξi,θ,a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' A feasible solution to LP (2) is said to be regular if it is not irregular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' It is easy to see that, given a regular feasible solution to LP (2), we can recover a feasible solution to Problem (1) with the same objective function value by simply letting pi,θ,a,ω = yi,θ,a,ω/ξi,θ,a for every i ∈ N, θ ∈ ˜Θn, a ∈ Ai,θi, and ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' However, the same is not true for irregular solutions, as the operation above is clearly ill defined in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Nevertheless, we show that, given any irregular feasible solution to LP (2), it is always possible to build a regular solution by only incurring in an arbitrarily small loss in objective function value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Formally: Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given an instance I := (N, Θ, Ω, A) of Bayesian principal-multi-agent problem and an irregular solution to LP (2) with value VAL, for any ǫ > 0, it is possible to recover a regular solution to LP (2) with value at least VAL − ǫ(n τ(|I|) + 1) in time polynomial in |I| and 1 ǫ, where τ is a function defined as per Lemma 6 and |I| denotes the size of instance I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Finally, we are ready to prove that solving LP (2) in place of Problem (1) allows us to recover in polynomial time a DSIC menu of randomized contracts that only incurs in an arbitrarily small loss with respect to the value SUP of the supremum of Problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Formally: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given an instance I := (N, Θ, Ω, A) of Bayesian principal-multi-agent problem and an optimal solution to LP (2), for any ǫ > 0, it is possible to recover a feasible solution to Problem (1) with value at least SUP−ǫ(n τ(|I|)+ 1) in time polynomial in |I| and 1 ǫ, where τ is a function defined as per Lemma 6 and |I| denotes the size of instance I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 14 ARXIV PREPRINT - FEBRUARY 1, 2023 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='4 Approximation Algorithm LP (2) features exponentially-many variables and polynomially-many constraints, and, thus, it can be solved in poly- nomial time by applying the ellipsoid method ot its dual, provided access to a suitable polynomial-time separation oracle for the constraints of the dual (Gr¨otschel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In this last section, we show that, despite an (exact) polynomial-time separation oracle may not be available in our setting, it is always possible to design a polynomial-time approximate separation oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This, together with some ad hoc modifications to the ellipsoid method, allows us to design the desired approximation algorithm for the problem of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Indeed, in instances that satisfy the FOSD condition and have IR-supermodular succinct rewards, it is possible to design an (exact) polynomial-time separation oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Instead, in instances having DR-submodular succinct rewards, this is not possible, and, thus, we need an approximate separation oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='15 We start by introducing a relaxation of LP (2) (see LP (3) below) and by showing that the two LPs are indeed equivalent (see Lemma 9 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Such a preliminary step allows us to obtain a dual LP which has additional constraints on its variables, which will be crucial in order to design a polynomial-time approximate separation oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The relaxation of LP (2), which is obtained by replacing the ‘=’ in Constraints (2e) with a ‘≤’, reads as follows: max � θ∈ ˜Θn λθ � a∈An,θ tθ,a Rθ,a − � i∈N � θ∈ ˜Θn λθ � a∈Ai,θi � ω∈Ω Fi,θi,a,ω yi,θ,a,ω s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (3a) � a∈An,θ:ai=a tθ,a ≤ ξi,θ,a ∀i ∈ N, ∀θ ∈ ˜Θn, ∀a ∈ Ai,θi (3b) Constraints (2b)—(2d) and (2f)—(2i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For every instance of Bayesian principal-multi-agent problem, LP (2) and LP (3) have the same optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, given a feasible solution to LP (3), it is always possible to recover in polynomial time a feasible solution to LP (2) having at least the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' By Lemma 9, we can solve LP (3) instead of LP (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The dual problem of LP (3) reads as follows:16 min � i∈N � θ∈Θ � θ−i∈ ˜Θn −i xi,θ,θ−i s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (4a) −1 � (θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' θ−i) ∈ ˜Θn� � θ′∈Θ yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ′ ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a + � θ′∈Θ: (θ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i)∈ ˜Θn � a′∈Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ′ ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a′ zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='(θ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a′ +di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i − 1 � (θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' θ−i) ∈ ˜Θn� xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i ≥ 0 ∀i ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ−i ∈ ˜Θn −i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀a ∈ Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ (4b) −yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ + � a′∈Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θi zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a′ ≥ 0 ∀i ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ ˜Θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀a ∈ Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ (4c) 1 � (θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' θ−i) ∈ ˜Θn� � θ′∈Θ yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ′ Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω − � θ′∈Θ: (θ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i)∈ ˜Θn � a′∈Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ′ Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='(θ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a′ ≥ −1 � (θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' θ−i) ∈ ˜Θn� λ(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i) Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω ∀i ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ−i ∈ ˜Θn −i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀a ∈ Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀ω ∈ Ω (4d) � i∈N yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ai ≥ λθ Rθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a ∀θ ∈ ˜Θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀a ∈ An,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ (4e) xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i ∀i ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ−i ∈ ˜Θn −i (4f) yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a ≥ 0 ∀i ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ ˜Θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀a ∈ Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θi (4g) zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a′ ≤ 0 ∀i ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ ˜Θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀a ∈ Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀a′ ∈ Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θi (4h) di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i ∀i ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ ∈ Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ∀θ−i ∈ ˜Θn −i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (4i) where xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i are dual variables that correspond to Constraints (2d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a to Constraints (2e),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a′ to Con- straints (2c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' while di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i to Constraints (2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 15Notice that the existence of an exact oracle for instances with DR-submodular rewards would contradict Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 16Notice that, in LP (4), we used 1 {·} to the denote the indicator function for the event written within curly braces 15 ARXIV PREPRINT - FEBRUARY 1, 2023 The dual LP (4) features polynomially-many variables and exponentially-many constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='17 Moreover, Con- straints (4e) are the only ones which are exponential in the size of the problem instance, since there is a group of such constraints for every tuple of agents’ actions a ∈ An,θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Thus, in order to have the ellipsoid method running in polynomial time on LP (4), it is sufficient to design a polynomial-time separation oracle for Constraints (4e), as the others can be checked one by one in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' As we show next, only an “approximate version” of such a separation oracle can be implemented in polynomial time, according to the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Definition 9 (Approximate separation oracle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given any α ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' an approximate separation oracle for Con- straints (4e) is a procedure Oα(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ·) which,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' given in input an instance I := (N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' A) of Bayesian principal- multi-agent problem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' a tuple of agents’ types θ ∈ ˜Θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' a vector w ∈ Rnℓ of weights—with wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a denoting the vector component corresponding to agent i ∈ N and action a ∈ Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θi—,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' and an additive error ǫ > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' returns a tuple of agents’ actions a ∈ An,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ such that: λθ Rθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a − � i∈N wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ai ≥ α λθ Rθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a′ − � i∈N wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a′ i − ǫ ∀a′ ∈ An,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' in time polynomial in |I|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' maxi∈N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a∈A |wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' and 1 ǫ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' where |I| denotes the size of instance I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Notice that, by letting each weight wi,a be equal to yi,θ,a for some feasible solution to LP (4), the problem solved in a call Oα(I, w, θ, ǫ) to the approximate separation oracle intuitively consists in finding the most violated constraint among Constraints (4e), up to a reward-multiplying approximation factor α and an additive error ǫ given as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Next, we show that it is possible to apply an ad hoc implementation of the ellipsoid method to LP (4), which, given access to an approximate separation oracle for Constraints (4e) as in Definition 9, returns a feasible solution to LP (4) that provides a desirable approximation of the optimal value of LP (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Such a procedure can be embedded in a suitable binary search scheme, resulting in a polynomial-time approximation algorithm for the principal’s optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given access to an approximate separation oracle Oα(·, ·, ·, ·) with α ∈ (0, 1], there exists an algorithm that, given any ρ > 0 and instance of Bayesian principal-multi-agent problem as input, returns a DSIC menu of randomized contracts with principal’s expected utility at least α RΓ − PΓ − ρ for every menu of randomized contracts Γ = {γθ}θ∈Θn, where RΓ ∈ [0, 1], respectively PΓ ∈ R+, denotes the expected reward, respectively the expected overall payment, of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, such an algorithm runs in time polynomial in the instance size and 1 ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We conclude the section by showing that the approximate separation oracle Oα(·, ·, ·, ·) can be implemented in poly- nomial time for two classes of Bayesian principal-multi-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This is the last step needed to fully specify the approximation algorithm introduced in Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In Bayesian principal-multi-agent problem instances that satisfy the FOSD condition and have IR-supermodular suc- cinct rewards, we are able to design a polynomial-time approximate separation oracle Oα(·, ·, ·, ·) with α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='18 Instead, in instances with DR-submodular succinct rewards, we sow to implement an oracle Oα(·, ·, ·, ·) with α = 1−1/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' These implementations work by solving suitably-defined problems that resemble non-Bayesian principal- multi-agent instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In particular, they have their same structure, while the rewards are scaled by a factor λθ and the values �Pi,a are substituted by the weights wi,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Formally, we get the following two results: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In Bayesian principal-multi-agent problem instances that (i) have succinct rewards specified by an IR- supermodular function and (ii) satisfy the FOSD condition, for any ρ > 0, the problem of computing an optimal menu of randomized contracts admits an algorithm returning a menu with principal’s expected utility at least OPT − ρ in time polynomial in the instance size and 1 ρ, where OPT is the value of the optimal principal’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In Bayesian principal-multi-agent problem instances with succinct rewards specified by a DR- submodular function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' the problem of computing an optimal menu of randomized contracts admits a polynomial-time approximation algorithm which,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' for any ǫ > 0 given as input,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' outputs a menu providing the principal with an expected utility at least of (1 − 1/e)RΓ − PΓ − ǫ for each menu of randomized contracts Γ = {γθ}θ∈Θn with high probability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' where RΓ ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' respectively PΓ ∈ R+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' denotes the expected reward,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' respectively the expected payment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' in contract p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 17We recall that the distribution λ is part of the problem instance given as input to our algorithm, and, thus, both |˜Θn| and |˜Θn −i| are polynomial quantities in the size of such instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 18Notice that, in Bayesian principal-multi-agent problem instances that satisfy the FOSD condition and have IR-supermodular succinct rewards, it is easy to adapt our results so as to show that there exists an exact separation oracle (thus getting rid of the additive error ǫ > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We decided to use an approximate separation oracle anyway, for ease of exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This choice does not detriment the final approximation guarantees of the algorithm (see Corollary 4), since we cannot get rid of the additive approximation ρ > 0 given that Problem (1) may not admit a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 16 ARXIV PREPRINT - FEBRUARY 1, 2023 Notice that Corollary 4 provides the same approximation guarantees of its corresponding result for non-Bayesian instances (see Theorem 6), while Corollary 3 matches those of its corresponding non-Bayesian result up to an additive error ρ > 0 (see Theorem 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 17 ARXIV PREPRINT - FEBRUARY 1, 2023 References Tal Alon, Paul D¨utting, and Inbal Talgam-Cohen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Contracts with Private Cost per Unit-of-Effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In Proceedings of the 22nd ACM Conference on Economics and Computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 52–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Chien-Ju Ho, Aleksandrs Slivkins, and Jennifer Wortman Vaughan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Adaptive contract design for crowdsourcing markets: Bandit algorithms for repeated principal-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Journal of Artificial Intelligence Research 55 (2016), 317–359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Ran Raz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' A parallel repetition theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 27, 3 (1998), 763–803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Alexander Schrijver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' A combinatorial algorithm minimizing submodular functions in strongly polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Journal of Combinatorial Theory, Series B 80, 2 (2000), 346–355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Alexander Schrijver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Combinatorial optimization: polyhedra and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Yoav Shoham and Kevin Leyton-Brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Multiagent systems: Algorithmic, game-theoretic, and logical founda- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Maxim Sviridenko, Jan Vondr´ak, and Justin Ward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Optimal approximation for submodular and supermodular optimization with bounded curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Mathematics of Operations Research 42, 4 (2017), 1197–1218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Steve Tadelis and Ilya Segal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Lectures in contract theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Lecture notes for UC Berkeley and Stanford University (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' David Zuckerman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Linear Degree Extractors and the Inapproximability of Max Clique and Chromatic Number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Theory of Computing 3, 6 (2007), 103–128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Lars Peter Østerdal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The mass transfer approach to multivariate discrete first order stochastic dom- inance: Direct proof and implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Journal of Mathematical Economics 46, 6 (2010), 1222–1228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='jmateco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='018 The Conferences at Barcelona, Milan, New Haven, San Diego and Tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 19 ARXIV PREPRINT - FEBRUARY 1, 2023 A Proofs Omitted from Section 3 Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' There always exists a base S∗ ∈ B(MI) of MI such that f I(S∗) = maxS∈II f I(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Let S ∈ II be any independent set of MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Clearly, by adding to S all the ground elements (i, a∅) ∈ GI i for i ∈ N \\ NS, we obtain a base S′ ∈ II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' By definition of the null action a∅, it holds �Pi,aS,i = �Pi,a∅ = 0 for every i ∈ N \\ NS, which implies f I(S) = f I(S′), since RaS = RaS′ given that aS = aS′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given an instance I := (N, Ω, A) of principal-multi-agent problem, the problem of computing a contract maximizing the principal’s expected utility can be reduced in polynomial time to solving maxS∈II f I(S) over the 1-partition matroid MI = ( � GI i � i∈N , II), where f I : 2GI → R is a set function such that, for every independent set S ∈ II, it holds: f I(S) := RaS − � i∈N �Pi,aS,i, where �Pi,aS,i = min p∈Pi,aS,i � ω∈Ω Fi,aS,i,ω pi,ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We prove the result by showing that, given any pair (p, a)—where p ∈ Rn×m + is a contract and a = (ai)i∈N ∈ ×i∈N A∗ i (p) is a tuple of IC agents’ actions recommended to the agents—, there exists a base S ∈ II of MI such that f I(S) is greater than or equal to the principal’s expected utility under (p, a), and, conversely, given any base S ∈ II there exists a pair (p, a) with principal’s expected utility f I(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This, together with Lemma 1, proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' From (p, a) to a base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Let the base S ∈ II be defined so that S := {(i, ai) : i ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, given that ai ∈ A∗ i (p) for all i ∈ N and by the definition of �Pi,aS,i, it holds: f I(S) = RaS − � i∈N �Pi,aS,i ≥ Ra − � i∈N � ω∈Ω Fi,ai,ω pi,ω = Ra − � i∈N Pi,ai, where the inequality holds since aS = a and the fact that, for every i ∈ N, the value �Pi,aS,i is defined as a minimum taken over the set Pi,aS,i, which contains the contract p given that ai ∈ A∗ i (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' From a base to (p, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given a base S ∈ II of the matroid MI, let (p, a) be such that a = (ai)i∈N satisfies (i, ai) ∈ S for all i ∈ N and p ∈ arg minp′∈Pi,aS,i � ω∈Ω Fi,aS,i,ω p′ i,ω for every i ∈ N (notice that such a contract can be built by defining the components pi,ω for ω ∈ Ω independently for each i ∈ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, it immediately follows from the definition of the function f I that the principal’s expected utility under (p, a) is equal to f I(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The problem of maximizing an ordered-supermodular function over a 1-partition matroid can be reduced in polynomial time to maximizing a supermodular function over a ring of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given a 1-partition matroid M := ({Gi}i∈[d] , I) and a function f : 2G → R that is ordered-supermodular, we show that maximizing f over M is equivalent to maximizing a suitably-defined supermodular function ˜f : R → R over a particular ring of sets R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The latter is defined by the family of all the sets S ⊆ G such that, if x ∈ S and x = π−1 i (j) for some i ∈ [d] and j ∈ [ki], then πi(l) ∈ S for all l ∈ [ki] : l < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, for every S ⊆ G, we let ˜f(S) := f(∧S), where ∧S denotes the set obtained by taking an element x ∈ S with maximal value of π−1 i (x) for each partition i ∈ [d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, it is sufficient to show that ˜f is supermodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Indeed, given two sets S, S′ ⊆ G, it holds: ˜f(S) + ˜f(S′) = f(∧S) + f(∧S′) ≤ f(∧(S ∪ S′)) + f(∧(S ∩ S′)) = ˜f(S ∪ S′) + ˜f(S ∩ S′), which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The problem of maximizing an ordered-supermodular function over a 1-partition matroid admits a polynomial-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The problem can be reduced in polynomial time to the maximization of a supermodular function defined over a ring of sets by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Such a problem is known to be solvable in polynomial time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', (Schrijver, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Bach, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 20 ARXIV PREPRINT - FEBRUARY 1, 2023 B Proof of Theorem 3 In order to prove the theorem, we employ a reduction from a promise problem associated with LABEL-COVER instances, whose definition follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Definition 10 (LABEL-COVER instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' An instance of LABEL-COVER is a tuple (G, Σ, Π): G := (U, V, E) is a bipartite graph defined by two disjoint sets of nodes U and V , connected by the edges in E ⊆ U × V , which are such that all the nodes in U have the same degree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Σ is a finite set of labels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' and Π := {Πe : Σ → Σ | e ∈ E} is a finite set of edge constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, a labeling of the graph G is a mapping π : U ∪ V → Σ that assigns a label to each vertex of G such that all the edge constraints are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Formally, a labeling π satisfies the constraint for an edge e = (u, v) ∈ E if it holds that π(v) = Πe(π(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The classical LABEL-COVER problem is the search problem of finding a valid labeling for a LABEL-COVER in- stance given as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In the following, we consider a different version of the problem, which is the promise problem associated with LABEL-COVER instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Definition 11 (GAP-LABEL-COVERc,s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For any pair of numbers 0 ≤ s ≤ c ≤ 1, we define GAP-LABEL- COVERc,s as the following promise problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Input: An instance (G, Σ, Π) of LABEL-COVER such that either one of the following is true: – there exists a labeling π that satisfies at least a fraction c of the edge constraints in Π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' – any labeling π satisfies less than a fraction s of the edge constraints in Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Output: Determine which of the above two cases hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' To prove Theorem 3, we use the following result due to Raz (1998) and Arora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Theorem 9 (Raz (1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Arora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (1998)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For any ǫ > 0, there exists a constant kǫ ∈ N that depends on ǫ such that the promise problem GAP-LABEL-COVER1,ǫ restricted to inputs (G, Σ, Π) with |Σ| = kǫ is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Now, we are ready to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given an approximation factor ρ > 0, we reduce from the problem GAP-LABEL-COVER1,ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Our construction is such that, if the LABEL-COVER instance admits a labeling that satisfies all the edge constraints, then the corresponding principal-multi-agent problem admits a contract providing the principal with an overall ex- pected utility of at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Otherwise, if at most a fraction ρ of the constraints are satisfied, then any contract provides the principal with an overall expected utility of at most ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Since ρ > 0 can be an arbitrarily small constant, this is sufficient to prove the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given an instance of GAP-LABEL-COVER1,ρ (G, Σ, Π) with a bipartite graph G = (U, V, E), we build a principal-multi-agent instance as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The set of agents includes an agent nv for every node v ∈ U ∪ V of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The outcome space has kρ dimensions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', Ω = Rkρ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Each agent nv, v ∈ V ∪ U has an action aσ for each label σ ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given an label σ, let ωσ ∈ Rkρ + be the outcome with ωσ σ = 1 and ωσ σ′ = 0 for each ω′ ̸= σ, where for ease of exposition we rename the set Σ as {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', kρ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For each agent nv and each action aσ, with σ ∈ Σ, cost cn,aσ = 0 and aσ induces the outcome ωσ deterministically, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', Fn,aσ,ωσ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Finally, the principal’s reward function g is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For each vector ω ∈ Ωn, g(ω) = � (v,u)∈E � σ∈Σ 1{ωnv,σ = 1 ∧ ωnu,Πe(σ) = 1}/|E|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' It is easy to see that the function is IR-supermodular in [0, 1]n|Σ| and hence for all the inducible outcomes ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='19 19It is easy to construct an arbitrary good approximation of g(·) that is IR-supermodular on all the domain Rn|Σ| + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For instance, we can set g(ω) = eM(ωnv,σ+ωnu,Πe(σ)−2)/|E| for an arbitrary large M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 21 ARXIV PREPRINT - FEBRUARY 1, 2023 Completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Suppose that the instance of GAP-LABEL-COVER1,ρ (G, Σ, Π) admits a labeling π : U ∪ V → Σ that satisfies all the edge constraints in Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Let us define a contract that recommends action aπ(v) for every node v ∈ U ∪ V , while all the payments are set to 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', pn,ω = 0 for each n ∈ N and ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Notice that the agents follow the recommendations since they are indifferent among all the actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' It is easy to see that the utility is 1 since for each edge (u, v), ωnv,π(v) = 1 and ωnu,Πe(π(u)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This concludes the first part of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Soundness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We show that, if the LABEL-COVER instance is such that every labeling π : U ∪ V → Σ satisfies at most a fraction ρ of the edge constraints in Π, then, in the corresponding principal-agent setting, any contract provides the principal with an expected utility at most ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Let ˆa be the tuple of action recommendations and recall that each action ˆanv, v ∈ V ∪ U induces deterministically an outcome ωnv ∈ {ωσ}σ∈Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' As a first step, notice that for each edge e = (u, v), � σ∈Σ 1(ωnv,σ = 1 ∧ ωnu,Πe(σ) = 1)/|E| is at most 1/|E| since there is exactly one σ such that ωnv,σ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Suppose by contradiction that there exists a contract with utility strictly larger than ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, there are strictly more than ρ|E| edges such that � σ∈Σ 1(ωnv,σ = 1 ∧ ωnu,Πe(σ) = 1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Consider the assignment that assign to each variable v ∈ V ∪ U the label σ such that ˆanv = aσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' It is easy to see that this assignment satisfies strictly more than a ρ fraction of the edges, reaching a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' C Proofs Omitted from Section 4 To prove the results in this section it will be useful to employ the definition of supermodularity for continuous func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Indeed, the properties introduced in Definition 2 are special cases of the classical submodularity and super- modularity properties which are usually considered in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Formally, by letting max{ω, ω}, respectively min{ω, ω′}, be the component-wise maximum, respectively minimum, between two given vectors ω, ω′ ∈ Rnq + , the following definition holds: Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' A reward function g : Rnq + → R is submodular if the following holds: g(ω) + g(ω′) ≥ g(max{ω, ω}) + g(min{ω, ω}) ∀ω, ω′ ∈ Rnq + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, a reward function g : Rnq + → R is supermodular if its opposite function −g is submodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' It is well known that any DR-submodular, respectively IR-supermodular, function is also submodular, respectively supermodular, but the converse is not true (Bian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In principal-multi-agent problems with succinct rewards that satisfy the FOSD condition, for every agent i ∈ N and pair aj, ak ∈ A of agent i’s actions such that j < k, there exists a collection of probability distributions µω ∈ ∆Ω−, one per outcome ω ∈ Ω, which are supported on the finite subset of the positive orthant Ω− := Rm + ∩ {ω − ω′ | ω, ω′ ∈ Ω} and satisfy the following equations: Fi,ak,ω = � ω′∈Ω Fi,aj,ω′ µω′ ω−ω′ ∀ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The proof follows from Theorem 1 in (Østerdal, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In particular, for every agent i ∈ N and action index j ∈ [ℓ − 1], given that the FOSD condition ensures that � ω∈Ω′ Fi,aj+1,ω ≤ � ω∈Ω′ Fi,aj,ω for all comprehensive sets Ω′ ⊆ Ω, Theorem 1 in (Østerdal, 2010) states that Fi,aj can be derived from Fi,aj+1 by means of a finite sequence of deteriorating bilateral transfers (of mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' These are operations which consist in moving probability mass from an outcome ω ∈ Ω to another outcome ω′ ∈ Ω such that ω′ ≤ ω, while maintaining the probability mass on outcomes that are different from ω and ω′ untouched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' As a result, given an agent i ∈ N and a pair aj, ak ∈ A of agent i’s actions such that j < k, it is easy to check that the probability Fi,ak,ω which Fi,ak places on an outcome ω ∈ Ω can be expressed as a suitable combinations of the probabilities Fi,aj,ω′ which Fi,aj,ω′ places on outcomes ω′ ∈ Ω such that ω′ ≤ ω (since all the bilateral transfers involved in the processes of turning Fi,ak into Fi,aj are deteriorating).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given an instance I := (N, Ω, A) of principal-multi-agent problem that (i) has succinct rewards specified by an IR-supermodular function and (ii) satisfies the FOSD condition, the set function f I defined over the 1-partition matroid MI = ( � GI i � i∈N , II) is ordered-supermodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In order to show the result, we prove that, for every pair of independent sets S, S′ ∈ II: f I(S ∧ S′) + f I(S ∨ S′) ≥ f I(S) + f I(S′), 22 ARXIV PREPRINT - FEBRUARY 1, 2023 where the partition-wise “maximum” ∧ and “minimum” ∨ are defined with respect to the bijective functions πi : [ki] → GI i (with ki = ℓ) constructed according to the (agent-dependent) ordering of the action set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In particular, for every i ∈ N and j ∈ [ℓ], it holds πi(j) = (i, aj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For ease of presentation, in the rest of the proof we let a1 := aS∧S′ and a2 := aS∨S′, so that a1,i, respectively a2,i, denotes the i-th component of a1, respectively a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' First, let us notice that � i∈N �Pi,a1,i +� i∈N �Pi,a2,i = � i∈N �Pi,aS,i +� i∈N �Pi,aS′,i, which holds since, by definition of partition-wise “maximum” ∧ and “minimum” ∨, for every agent i ∈ N the pair of actions a1,i, a2,i exactly coincides (up to ordering) with aS,i, aS′,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Thus, given the definition of f I (see Theorem 1), in order to prove the result it is sufficient to prove that Ra1 + Ra2 ≥ RaS + RaS′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' By definition of a1 and a2, we have that π−1 i (i, a1,i) ≥ π−1 i (i, a2,i) for every i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, thanks to Lemma 2 and how actions are ordered, for every agent i ∈ N, there exists a collection of probability distributions µi,ω ∈ ∆Ω−, one per outcome ω ∈ Ω, such that Fi,a1,i,ω = � ω′∈Ω Fi,a2,i,ω′µi,ω′ ω−ω′, where we recall that the µi,ω are the probability distributions that allow to turn Fi,a2,i into Fi,a1,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, notice that, whenever a1,i = a2,i, it holds µi,ω 0 = 1 for every ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' we can write: Ra1 = � ω∈Ωn �� i∈N Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωi � g(ω) = � ω∈Ω � i∈N � � ω′∈Ω Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′µi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ ω−ω′ � g(ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' RaS = � ω∈Ωn �� i∈N Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='aS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωi � g(ω) = � ω∈Ωn \uf8eb \uf8ec \uf8ed � i∈N: aS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i=a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i � ω′∈Ω Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′µω′ ω−ω′ \uf8f6 \uf8f7 \uf8f8 \uf8eb \uf8ec \uf8ec \uf8ed � i∈N: aS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i̸=a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωi \uf8f6 \uf8f7 \uf8f7 \uf8f8 g(ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' and RaS′ = � ω∈Ωn �� i∈N Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='aS′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωi � g(ω) = � ω∈Ωn \uf8eb \uf8ec \uf8ec \uf8ed � i∈N: aS′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i=a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i � ω′∈Ω Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′µω′ ω−ω′ \uf8f6 \uf8f7 \uf8f7 \uf8f8 \uf8eb \uf8ec \uf8ec \uf8ed � i∈N: aS′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i̸=a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωi \uf8f6 \uf8f7 \uf8f7 \uf8f8 g(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In the following, for ease of presentation, given a pair of tuples of agents’ outcomes ω, ω′ ∈ Ωn such that ωi ≥ ω′ i for every i ∈ N, we denote by ωω,ω′ 1 , ωω,ω′ 2 ∈ Ωn another pair of tuples of agents’ outcomes, which depend on ω, ω′ and are defined as follows: if agent i ∈ N is such that a1,i = aS,i and a2,i = aS′,i, then it holds ωω,ω′ 1,i = ωi and ωω,ω′ 2,i = ω′ i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' if agent i ∈ N is such that a1,i = aS′,i and a2,i = aS,i, then it holds ωω,ω′ 1,i = ω′ i and ωω,ω′ 2,i = ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Notice that, as it is easy to check, it holds ω = ωω,ω′ 1 ∧ ωω,ω′ 2 and ω = ωω,ω′ 1 ∨ ωω,ω′ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Thus, it is also the case that g(ω) + g(ω′) ≤ g(ωω,ω′ 1 ) + g(ωω,ω′ 2 ), sine the reward function g is IR-supermodular and hence supermodular (see Definition 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Let N1 := {i ∈ N : aS,i = a1,i}, and N2 := {i ∈ N : aS′,i = a1,i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In order to conclude the proof,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 23 ARXIV PREPRINT - FEBRUARY 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2023 we show that the following holds: Ra1 + Ra2 = � ω∈Ωn �� i∈N Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωi � g(ω) + � ω∈Ωn �� i∈N Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωi � g(ω) (5a) = � ω∈Ωn �� i∈N � ω′∈Ω Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′µi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ ωi−ω′ � g(ω) + � ω∈Ωn �� i∈N Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωi � g(ω) (5b) = � ω∈Ωn � ω′∈Ωn �� i∈N Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i � �� i∈N µi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i ωi−ω′ i � g(ω) + � ω∈Ωn �� i∈N Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωi � g(ω) (5c) = � ω′∈Ωn �� i∈N Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i � � � ω∈Ωn �� i∈N µi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i ωi−ω′ i � g(ω) + g(ω′) � (5d) = � ω′∈Ωn �� i∈N Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i � � � ω∈Ωn �� i∈N µi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i ωi−ω′ i � g(ω) + � ω∈Ωn �� i∈N µi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i ωi−ω′ i � g(ω′) � (5e) ≥ � ω′∈Ωn �� i∈N Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i � � ω∈Ωn �� i∈N µi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i ωi−ω′ i � � g(ωω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ 1 ) + g(ωω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ 2 ) � (5f) = � ω′∈Ωn �� i∈N Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i � \uf8ee \uf8ef\uf8ef\uf8f0 � ω∈Ωn: ωi=ω′ i∀i∈N2 � i∈N: aS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i=a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i µi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i ωi−ω′ i g(ωω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ 1 ) + � ω∈Ωn: ωi=ω′ i∀i∈N1 � i∈N: aS′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i=a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i µi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i ωi−ω′ i g(ωω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ 2 ) \uf8f9 \uf8fa\uf8fa\uf8fb (5g) = � ω∈Ωn \uf8eb \uf8ec \uf8ec \uf8ed � i∈N: aS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i̸=a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωi \uf8f6 \uf8f7 \uf8f7 \uf8f8 � ω′∈Ωn: ωi=ω′ i∀i∈N2 � i∈N: aS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i=a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i µi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i ωi−ω′ i g(ωω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ 1 ) + � ω′∈Ωn \uf8eb \uf8ec \uf8ec \uf8ed � i∈N: aS′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i̸=a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i \uf8f6 \uf8f7 \uf8f7 \uf8f8 � ω∈Ωn: ωi=ω′ i∀i∈N1 � i∈N: aS′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i=a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i µi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i ωi−ω′ i g(ωω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ 2 ) (5h) = � ω∈Ωn \uf8eb \uf8ec \uf8ec \uf8ed � i∈N: aS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i̸=a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωi \uf8f6 \uf8f7 \uf8f7 \uf8f8 � ω′∈Ωn: ωi=ω′ i∀i∈N2 � i∈N: aS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i=a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i µi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i ωi−ω′ i g(ω) + � ω∈Ωn \uf8eb \uf8ec \uf8ec \uf8ed � i∈N: aS′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i̸=a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωi \uf8f6 \uf8f7 \uf8f7 \uf8f8 � ω′∈Ωn: ωi=ω′ i∀i∈N1 � i∈N: aS′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i=a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i µi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i ωi−ω′ i g(ω) (5i) = � ω∈Ωn \uf8eb \uf8ec \uf8ec \uf8ed � i∈N: aS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i̸=a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωi \uf8f6 \uf8f7 \uf8f7 \uf8f8 \uf8eb \uf8ec \uf8ed � i∈N: aS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i=a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i � ω′∈Ω Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i µi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i ωi−ω′ \uf8f6 \uf8f7 \uf8f8 g(ω) + � ω∈Ωn \uf8eb \uf8ec \uf8ec \uf8ed � i∈N: aS′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i̸=a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωi \uf8f6 \uf8f7 \uf8f7 \uf8f8 \uf8eb \uf8ec \uf8ec \uf8ed � i∈N: aS′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i=a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i � ω′∈Ω Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i µi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i ωi−ω′ \uf8f6 \uf8f7 \uf8f7 \uf8f8 g(ω) (5j) = RaS + RaS′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (5k) 24 ARXIV PREPRINT - FEBRUARY 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 2023 where Equation (5b) follows from the definition of Ra1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Equation (5e) comes from the fact that � ω∈Ωn � i∈N µi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ i ωi−ω′ i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Equation (5f) holds since g(ω) + g(ω′) ≤ g(ωω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ 1 ) + g(ωω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ 2 ) by supermodular- ity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Equation (5g) follows from ωω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ 1 = ωω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′′ 1 whenever ω′ i = ω′′ i for all i ∈ N1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ωω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ 2 = ωω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′′ 2 whenever ω′ i = ω′′ i for all i ∈ N2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' and � ω∈Ω µi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ ωi−ω′ = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Equation (5i) comes from ωω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ 1 = ωω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′′ 1 whenever ω′ i = ω′′ i for all i ∈ N1 and ωω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′ 2 = ωω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω′′ 2 whenever ω′ i = ω′′ i for all i ∈ N2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' while Equation (5k) follows from the definition of RaS+ RaS′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For principal-multi-agent problem instances that (i) have succinct rewards specified by an IR- supermodular function and (ii) satisfy the FOSD condition, the problem of computing an optimal contract admits a polynomial-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' By Theorem 1, computing a utility-maximizing contract in a principal-multi-agent problem instance I := (N, Ω, A) can be reduced in polynomial time to the problem of maximizing a suitably-defined set function f I over a particular 1-partition matroid MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, by Lemma 3, the function f I is ordered-supermodular whenever the principal’s rewards are specified by an IR-supermodular function and the FOSD condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Hence, Corollary 1 immediately provides a polynomial-time algorithm for finding a utility-maximizing contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' D Proof of Theorem 5 To prove the theorem, we employ a reduction from a promise problem related to the problem of finding large indepen- dent sets in graphs, whose definition follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Definition 13 (GAP-ISα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For every α ∈ [0, 1], we define GAP-ISα as the following promise problem: Input: An undirected graph G = (V, E) such that either one of the following is true: – there exists an independent set (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', a subset of vertices such that there is no edge connecting two of them) of size at least |V |1−α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' – all the independent sets have size at most |V |α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Output: Determine which of the above two cases hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' GAP-ISα is known to be NP-hard for any α > 0 (H˚astad, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Zuckerman, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Now, we are ready to prove of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given a constant α > 0, we reduce from the problem GAP-ISα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Our construction is such that, if the GAP-ISα instance G = (V, E) admits an independent set of size at least |V |1−α, then the corresponding contract design problem admits a solution providing the principal with an overall expected utility at least of δ|V |1−α, where δ will be defined in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Otherwise, if all the independent sets have size at most |V |α, then any contract provides the principal with an overall expected utility at most of δ|V |α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, we will see that in the multi-agent principal-agent problem in our reduction it holds n = |V |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Since GAP-ISα is NP-hard for each constant α > 0 this is sufficient to prove the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given an instance of GAP-ISα G = (V, E), we build an instance of the multi-agent principal-agent problem as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For each vertex v ∈ V , there exists an agent nv with actions a1 and a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The outcome space is given by R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, for each agent v ∈ V action a1 induces deterministically outcome ω1 = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', Fnv,a1,ω1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, action a1 has cost 1 − δ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', cnv,a1 = 1 − δ, where δ = 1 |V |2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For each agent v ∈ V action a0 induces deterministically outcome ω0 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', Fnv,a0mω0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, action a0 has cost 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', cnv,a0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given a node v ∈ V , let kv ≤ |V | be the degree of node v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Finally, the utility function g is defined as g(ω) = � (u,v)∈E max{ 1 ku ωu, 1 kv ωv}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Notice that the function is DR-submodular since it is the sum of DR-submodular functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Indeed, given and edge e = (u, v), max{ 1 ku ωu, 1 kv ωv} is the maximum of two linear functions and hence is DR-submodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 25 ARXIV PREPRINT - FEBRUARY 1, 2023 Completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Suppose that there exists an independent set V ∗ ⊆ V of G with size at least |V |1−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We can build a contracts (p, a), p ∈ Rn×m + , a ∈ An such that for each v ∈ V ∗, it holds pnv,ω1 = 1 − δ, while all the other payments are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Finally, we recommend to all the agent nv, v ∈ V ∗, to play a1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', anv = a1, and to all the agents nv, v ∈ V \\ V ∗ to play a0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', anv = a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' It is easy to see that the action profile a is such that ai is IC under p for each agent i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The total reward is � (u,v)∈E max{ 1 ku ωu, 1 kv ωv} = � v∈V ∗ � u∈V :(u,v)∈E max{ 1 ku ωu, 1 kv ωv} = � v∈V ∗ � u∈V :(u,v)∈E 1 kv 1 = |V ∗||kv| 1 kv = |V ∗|, where the second inequality holds since for each v ∈ V ∗ we have that ωv = 1 and ωu = 0 for each u : (u, v) ∈ V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, the total payment is given by � v∈V ∗(1 − δ) = |V ∗|(1 − δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Thus, the total principal’s utility is given by |V ∗| − (1 − δ)|V ∗| = δ|V ∗|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Soundness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We prove that, if all the independent sets of G have size at most |V |α, then the principal’s expected utility is at most δ|V |α for any contract p ∈ R+n × m, a ∈ An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' First, we show that if the contract incentivizes two agents nu and nv with (u, v) ∈ E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', u and v are adjacent vertexes, to play action a1, then the principal’s utility is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Let ¯V be the set of nodes relative to agents incentivized to play a1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', the set of i ∈ N such that ai = a1, and ¯E be the set of edges connecting two nodes in ¯V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, the principal’s reward is at most � (u,v)∈E max{ 1 ku ωu, 1 kv ωv} = � (u,v)∈E\\ ¯ E max{ 1 ku ωu, 1 kv ωv} + � (u,v)∈ ¯ E max{ 1 ku ωu, 1 kv ωv} = � v∈ ¯V :(u,v)∈E\\ ¯ E 1 kv + � (u,v)∈ ¯ E [ 1 ku ωu + 1 kv ωv − 1/|V |] = � v∈ ¯V :(u,v)∈E\\ ¯ E 1 kv + � (u,v)∈ ¯ E [ 1 ku ωu + 1 kv ωv] − 1/|V | = � v∈ ¯V :(u,v)∈E\\ ¯ E 1 kv + � v∈ ¯V :(u,v)∈ ¯ E 1 kv − 1/|V | = � v∈ ¯V � u∈V :(u,v)∈E 1 kv − 1/|V | ≤ | ¯V | − 1/|V |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' At the same time the payment is at least (1 − δ)| ¯V | since for each agent nv, v ∈ ¯V it holds pnv,ω1 ≥ 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Hence, the principal’s utility is at most | ¯V | − 1/|V | − (1 − δ)| ¯V | = δ| ¯V | − 1/|V | < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Hence, in any contract with positive utility there are not two agents nv, nu relative to adjacent vertexes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', such that (v, u) ∈ E playing action a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Since all the independent sets has size at most |V |α, this implies that | ¯V | ≤ |V |α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, the reward of the contract is given by � v∈ ¯V � u:(u,v)∈E 1 kv 1 = | ¯V |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, the payment is at least (1 − δ)| ¯V | since for each agent nv, v ∈ ¯V it holds pnv,ω1 ≥ 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' However, the principal’s utility is at most | ¯V | − (1 − δ)| ¯V | = δ| ¯V | ≤ δ|V |α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' E Proofs Omitted from Section 5 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given an instance I := (N, Ω, A) of principal-multi-agent problem with succinct rewards specified by a DR-submodular function, the extended set function f I (see Definition 7) can be defined as f I(S) := fI(S) + lI(S) for every S ⊆ GI, where fI : 2GI → R+ is a monotone-increasing submodular function and lI : 2GI → R is a linear function, both defined over the 1-partition matroid MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 26 ARXIV PREPRINT - FEBRUARY 1, 2023 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' By letting fI : 2GI → R+ and lI : 2GI → R be defined so that fI(S) := RS and lI(S) := � (i,a)∈S �Pi,a for every S ⊆ GI, in order to prove the statement it is sufficient to show that fI is a monotone-increasing submodular function (notice that lI is linear by definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' It is easy to check that fI is monotone-increasing, since the reward function g is increasing by assumption (see As- sumption 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Thus, we are left to show that fI is also submodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In the following, for ease of presentation, given an agent i ∈ N and an outcome ω ∈ Ω, we let ωi,ω ∈ Ωn be the tuple of agents’ outcomes such that ωi,ω i = ω and ωi,ω j = ω∅ for all j ∈ N : j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, given two tuples ω, ω′ ∈ Ωn, we let ω + ω′ be the tuple whose i-th outcome is ωi + ω′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In order to prove that fI is submodular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' we need to show that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' for any two subsets S ⊂ S′ ⊆ GI and element (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' a) ∈ GI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' it holds fI(S ∪ {(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' a)}) − fI(S) ≤ fI(S′ ∪ {(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' a)}) − fI(S′): fI(S′ ∪ {(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' a)}) − fI(S′) = RS∪{(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a)} − RS′ = � ω∈˜Ωn rω � j∈N Fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='S′∪{(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωj − � ˜ω∈Ωn rω � j∈N Fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='S′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωj = � ω∈˜Ωn � ω′∈˜Ωn ��� ω∈Ω rω+ω′+ωi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω \uf8eb \uf8ed � j∈N Fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωj \uf8f6 \uf8f8 \uf8eb \uf8ed � j∈N Fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='S′\\S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωj \uf8f6 \uf8f8 − � ω∈˜Ωn � ω′∈˜Ωn rω+ω′ \uf8eb \uf8ed � j∈N Fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωj \uf8f6 \uf8f8 \uf8eb \uf8ed � j∈N Fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='S′\\S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωj \uf8f6 \uf8f8 = � ω∈˜Ωn � ω′∈˜Ωn � ω∈Ω Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω \uf8eb \uf8ed � j∈N Fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωj \uf8f6 \uf8f8 \uf8eb \uf8ed � j∈N Fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='S′\\S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωj \uf8f6 \uf8f8 (rω+ω′+ωi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω − rω+ω′) ≤ � ω∈˜Ωn � ω′∈˜Ωn � ω∈Ω Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω \uf8eb \uf8ed � j∈N Fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωj \uf8f6 \uf8f8 \uf8eb \uf8ed � j∈N Fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='S′\\S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωj \uf8f6 \uf8f8 (rω+ωi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω − rω) = � ω∈˜Ωn � ω∈Ω Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω \uf8eb \uf8ed � j∈N Fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωj \uf8f6 \uf8f8 � rω+1i(¯ω) − rω � = � ω∈˜Ωn � ω∈Ω Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω \uf8eb \uf8ed � j∈N Fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωj \uf8f6 \uf8f8 rω+ωi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ω − � ω∈˜Ωn \uf8eb \uf8ed � j∈N Fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωj \uf8f6 \uf8f8 rω = � ω∈˜Ωn rω � j∈N Fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='S∪{(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωj} − � ω∈˜Ωn rω � j∈N Fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ωj = fI(S ∪ {(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' a)}) − fI(S),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' where the inequality hold by DR-submodularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In principal-multi-agent problems with succinct rewards specified by a DR-submodular function, the problem of computing an optimal contract admits a polynomial-time approximation algorithm that, for any ǫ > 0 given as input, outputs a contract with principal’s expected utility at least (1 − 1/e)R(p,a∗) − P(p,a∗) − ǫ for any contract (p, a∗) with high probability, where R(p,a∗) ∈ [0, 1], respectively P(p,a∗) ∈ R+, denotes the expected reward, respectively payment, under (p, a∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The result easily follows by noticing that, thanks to Lemma 4, the problem is a specific case of the ones studied in (Sviridenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In particular, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='1 in (Sviridenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', 2017) shows that there exists a polynomial-time algorithm that, given as input an ǫ > 0, a matroid M := (G, I), a monotone-increasing submodular function f : 2G → R+, and a linear function l : 2G → R, outputs an independent set S ∈ I satisfying f(S) + l(S) ≥ (1 − 1/e)f(S′) + l(S′) − ǫˆv for every S′ ∈ I with high probability, where, for ease of presentation, we let ˆv := max{maxx∈G(f({x}), maxx∈G |l({x})|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 27 ARXIV PREPRINT - FEBRUARY 1, 2023 It is easy to see that, by definition of reward function, it holds max(i,a)∈GI fI({(i, a)}) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, it is always possible to build a matroid which is equivalent (for our purposes) to MI and satisfies max(i,a)∈GI |lI({(i, a)})| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' To do that, let ˜GI ⊆ GI be the set of elements (i, a) such that �Pi,a > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, any independent set including an element of ˜GI cannot be optimal, since it has negative value (recall that the values of fI are in [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Hence, we can optimize over the matroid that only includes the elements in GI \\ ˜GI, so that we get ˆv ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' F Proofs Omitted from Section 6 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given an instance I := (N, Θ, Ω, A) of Bayesian principal-multi-agent problem and a DSIC menu of randomized contracts, there always exists another DSIC menu of randomized contracts Γ = (γθ)θ∈Θn with at least the same principal’s expected utility such that, for every i ∈ N and θ ∈ Θn, it holds that ��� pi | p ∈ supp(γθ) ∧ p ∈ Pi,θi,a��� ≤ 1 for all a ∈ A, where pi ∈ Rm + denotes the i-th row of matrix p (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', the agent i’s payment scheme under contract p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Consider a menu of randomized contract ˆγθ for each θ ∈ Θn such that given a ˆθ ∈ Θn and i ∈ N, there exists two contracts p, p′ ∈ supp(γˆθ) such that pi ̸= p′ i and {pi, p′ i} ⊆ Pi,ˆθi,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Let ¯p be such that ¯pj = pj for each j ̸= i and ¯pi = ˆγˆθ pipi + ˆγˆθ p′ ip′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, let ¯p′ be such that ¯p′ j = p′ j for each j ̸= i and ¯p′ i = ˆγˆθ pipi + ˆγˆθ p′ ip′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We build a DSIC menu of randomized contracts γθ for each θ ∈ Θn with at least the same principal’s utility such that γθ = ˆγθ for each θ ̸= ˆθ, γˆθ ¯p = ˆγˆθ p + ˆγˆθ ¯p , γˆθ ¯p′ = ˆγˆθ p′ + ˆγˆθ ¯p′, γˆθ ˆp = ˆγˆθ ˆp for each p /∈ {¯p, ¯p′, p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' It is easy to see that γθ satisfies ��� pi | p ∈ supp(γθ) ∧ p ∈ Pi,θi,a��� < ��� pi | p ∈ supp(ˆγθ) ∧ p ∈ Pi,θi,a��� − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, γθ for each θ ∈ Θn provides the same utility of ˆγθ for each θ ∈ Θn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' To conclude the proof we need to proof that γθ for each θ ∈ Θn is DSIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Following an analysis similar to the one in Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (2022a) we can show that replacing the marginal contracts pi and p′ i with the weighted combination ¯pi the DSIC constraint continue to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Applying this operation until such two contracts does not exist is sufficient to prove the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' There exists a function τ : N → R such that τ(x) is O(2poly(x))—with poly(x) being a polynomial in x—and, for every instance I := (N, Θ, Ω, A) of Bayesian principal-multi-agent problem, agent i ∈ N, type θ ∈ Θ, and inducible action a ∈ Ai,θ, there exists a contract p ∈ Rn×m + such that a ∈ A∗ i,θ(p) and pi,ω ≤ τ(|I|) for all ω ∈ Ω, where |I| is the size of instance I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='20 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For every I := (N, Θ, Ω, A), agent i ∈ N, type θ ∈ Θ, and inducible action a ∈ Ai,θ, the set Pi,θ,a of contracts under which action a is IC can be defined by means of a system of linear inequalities such that its number of variables, its number of inequalities, and the size of the binary representation of its coefficients can all be bounded by polynomials in |I|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Hence, given that Pi,θ,a cannot be empty (otherwise a ∈ Ai,θ would be contradicted), there must exist a contract p ∈ Pi,θ,a such that, for every outcome ω ∈ Ω, the payment pi,ω is upper bounded by a O(2poly(|I|)) term, where poly(|I|) is a polynomial in the size |I| (in terms of number of bits) of instance I (this easily follows from standard LP arguments, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', (Bertsimas and Tsitsiklis, 1997)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The result is readily proved by choosing a suitable function τ : N → R so that such upper bound holds for every instance I, agent i ∈ N, type θ ∈ Θ, and inducible action a ∈ Ai,θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For every instance of Bayesian principal-multi-agent problem, it holds LP ≥ SUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 20In the rest of the section, we always assume that the size of a problem instance is expressed in terms of number of bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 28 ARXIV PREPRINT - FEBRUARY 1, 2023 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' To prove the result, we show that, given any feasible solution to Problem (1), it is possible to recover a feasible solution to LP (2) having the same objective function value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Let (tθ,a, ξi,θ,a, pi,θ,a,ω) be a feasible solution to Problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, we define a solution to LP (2) by letting yi,θ,a,ω = ξi,θ,a pi,θ,a,ω for every agent i ∈ N, tuple of agents’ types θ ∈ ˜Θn, inducible action a ∈ Ai,θi, and outcome ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Additionally, all the variables that also appear in Problem (1) keep their values, while variables γi,θ,θ,,a are defined so that they are equal to their corresponding terms in the sums appearing in the right-had sides of Constraints (1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' It is immediate to see that the solution defined above is indeed feasible for LP (2), after noticing that, for every i ∈ N, θ ∈ ˜Θn, and action a ∈ A which is not inducible for an agent i of type θi, it holds that ξi,θ,a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Indeed, since Constraints (1b) hold, if ξi,θ,a > 0 then there exists a contract p ∈ Rn×m + under which action a is IC for an agent i of type θi, and, thus, a ∈ Ai,θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' It is easy to check that the feasible solution to LP (2) defined above has exactly the same objective function value as its corresponding feasible solution to Problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Thus, the result is readily proved by observing that the objective functions of Problem (1) and LP (2) are continuous and, for any ǫ > 0, there always exists a feasible solution to Problem (1) with value at least SUP − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given an instance I := (N, Θ, Ω, A) of Bayesian principal-multi-agent problem and an irregular solution to LP (2) with value VAL, for any ǫ > 0, it is possible to recover a regular solution to LP (2) with value at least VAL − ǫ(n τ(|I|) + 1) in time polynomial in |I| and 1 ǫ, where τ is a function defined as per Lemma 6 and |I| denotes the size of instance I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Let (tθ,a, ξi,θ,a, yi,θ,a,ω, γi,θ,θ,,a) be a feasible solution to LP (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, let us define W as the set of tuples w = (i, θ, a) such that yi,θ,a,ω > 0 and ξi,θ,a = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', the tuples of indexes identifying the pairs of variables that do not meet regularity conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' As a first step, we show that, for every tuple w = (i, θ, a) ∈ W, it is possible to build a feasible solution to LP (2), which we refer to as � tw θ,a, ξw i,θ,a, yw i,θ,a,ω, γw i,θ,θ,,a � for clarity of exposition, such that its corresponding DSIC menu of randomized contracts always recommends action a with probability 1 to an agent i that truthfully reports their type to be θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Since a ∈ Ai,θi thanks to how LP (2) is constructed, Lemma 6 says that there exists a contract pw ∈ Rn×m + (depending on the tuple w = (i, θ, a)) such that a ∈ A∗ i,θi(pw) and pw i,ω ≤ τ(|I|) for all ω ∈ Ω, where τ : N → R is a suitably-defined function such that τ(x) is O(2poly(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, let us define yw i,θ,a,ω = pw i,ω for all ω ∈ Ω, while ξw i,θ,a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Additionally, for every θ′ ∈ ˜Θn and j ∈ N such that (θ′, j) ̸= (θ, i), by letting a′ ∈ Aj,θ′ j be any action that is inducible for an agent j of type θ′ j, we define ξw i,θ′,a′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Finally, we let tw θ,a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' It is easy to check that, by suitably defining all the unspecified variables, the solution � tw θ,a, ξw i,θ,a, yw i,θ,a,ω, γw i,θ,θ,,a � is feasible for LP (2) and it has value at least −n τ(|I|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In conclusion, for any ǫ > 0, let us consider a solution � t′ θ,a, ξ′ i,θ,a, y′ i,θ,a,ω, γ′ i,θ,θ,,a � to LP (2) whose components are defined as follows (by applying operations component wise): (1 − ǫ) (tθ,a, ξi,θ,a, yi,θ,a,ω, γi,θ,θ,,a) + � w∈W ǫ |W| � tw θ,a, ξw i,θ,a, yw i,θ,a,ω, γw i,θ,θ,,a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' It is easy to see that such a solution is feasible for LP (2), it is regular, and its objective function value is at least VAL − ǫ(n τ(|I|) + 1), where VAL is the value of the original (irregular) solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, it can be computed in time polynomial in |I| and 1 ǫ, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given an instance I := (N, Θ, Ω, A) of Bayesian principal-multi-agent problem and an optimal solution to LP (2), for any ǫ > 0, it is possible to recover a feasible solution to Problem (1) with value at least SUP−ǫ(n τ(|I|)+ 1) in time polynomial in |I| and 1 ǫ, where τ is a function defined as per Lemma 6 and |I| denotes the size of instance I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' First, let us recall that, given any regular feasible solution (tθ,a, ξi,θ,a, yi,θ,a,ω, γi,θ,θ,,a) to LP (2), it is suffi- cient to set pi,θ,a,ω = yi,θ,a,ω/ξi,θ,a for every i ∈ N, θ ∈ ˜Θn, a ∈ Ai,θi, and ω ∈ Ω, in order to recover a feasible solution to Problem (1) having the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Thus, if the given optimal solution to LP (2) is regular, then the result immediately follows by Lemma 7, since LP = SUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Instead, if the given optimal solution is irregular, by applying Lemma 8 we can recover a regular solution to LP (1) with value at least LP − ǫ(n τ(|I|) + 1) in time polynomial in |I| and 1 ǫ, and from that we can easily obtain a feasible solution to Problem (1) with value at least SUP − ǫ(n τ(|I|) + 1) (using the bound in Lemma 7), proving the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 29 ARXIV PREPRINT - FEBRUARY 1, 2023 Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For every instance of Bayesian principal-multi-agent problem, LP (2) and LP (3) have the same optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, given a feasible solution to LP (3), it is always possible to recover in polynomial time a feasible solution to LP (2) having at least the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Since LP (3) is a relaxation of LP (2), in order to prove the statement it is sufficient to show that, given a feasible solution to LP (3), it is possible to build a feasible solution to LP (2) having at least the same value, in time polynomial in the size of the instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Let (tθ,a, ξi,θ,a, yi,θ,a,ω, γi,θ,θ,,a) be a feasible solution to LP (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' As a first step, we show that, for every tuple of agents’ types θ ∈ ˜Θn, it is possible to compute new values for (some of the) variables tθ,a so as to obtain a new set of variables ˆtθ,a for a ∈ An,θ such that: (i) � a∈An,θ:ai=a ˆtθ,a = ξi,θ,a − � a:ai=a tθ,a for every i ∈ N and a ∈ Ai,θi, (ii) ˆtθ,a ≥ 0 for all a ∈ An,θ, and (iii) the new values can be computed in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' For ease of presentation, for every tuple of agents’ types θ ∈ ˜Θn, agent i ∈ N, and action a ∈ Ai,θi, let δi,θ,a := ξi,θi,a − � a∈An,θ:ai=a tθ,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, let δθ := 1 − � a∈An,θ tθ,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, for every agent i ∈ N, it holds: � a∈Ai,θi δi,θ,a = � a∈Ai,θi ξi,θi,a − � a∈Ai,θi � a∈An,θ:ai=a tθ,a = 1 − � a∈An,θ tθ,a = δθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Now, let ¯tθ,a or a ∈ An,θ be variable values identifying a probability distribution over action profiles in An,θ having marginal probabilities equal to δi,θ,a/δθ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=', for every i ∈ N and a ∈ Ai,θi, it holds � a∈An,θ:ai=a ¯tθ,a = δi,θ,a/δθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Notice that, since � a∈Ai,θi δi,θ,a = δθ for every i ∈ N, the marginal probabilities are well defined and the (joint) probability distribution exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, it is easy to see that such values ¯tθ,a can be computed in polynomial time, since there always exists a probability distribution as desired having a polynomially-sized support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, let us define ˆtθ,a = δθ¯tθ,a for every a ∈ An,θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Notice that such values satisfy all the conditions (i)–(iii), since � a:ai=a ˆtθ,a = δθ � a:ai=a ¯tθ,a = δi,θ,a = ξi,θ,a − � a:ai=a tθ,a, and ˆtθ,a ≥ 0 for all a ∈ An,θ, as the values ¯tθ,a identify a probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Now, let us consider a new solution to LP (3), namely (t′ θ,a, ξi,θ,a, yi,θ,a,ω, γi,θ,θ,,a), which is such that, for every θ ∈ ˜Θn and a ∈ An,θ, it holds t′ θ,a = tθ,a + ˆtθ,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In the rest of the proof, we show that the solution defined above is feasible for LP (2) and it has at least the same objective function value as the original feasible solution to LP (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' First, it is easy to check that the objective value does not decrease, since each t′ θ,a increases its value with respect to tθ,a and such variables appear with non-negative coefficients in the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Second, for every agent i ∈ N, tuple of agents’ types θ ∈ ˜Θn, and action a ∈ Aiθi, it holds: � a∈An,θ:ai=a t′ θ,a = � a∈An,θ:ai=a � tθ,a + ˆtθ,a � = � a∈An,θ:ai=a tθ,a + ξi,θ,a − � a∈An,θ:ai=a tθ,a = ξi,θ,a, where the second-to-last equality comes from condition (i) on ˆtθ,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Thus, such a solution is also feasible for LP (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, it is easy to see that it can be computed in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Given access to an approximate separation oracle Oα(·, ·, ·, ·) with α ∈ (0, 1], there exists an algorithm that, given any ρ > 0 and instance of Bayesian principal-multi-agent problem as input, returns a DSIC menu of randomized contracts with principal’s expected utility at least α RΓ − PΓ − ρ for every menu of randomized contracts Γ = {γθ}θ∈Θn, where RΓ ∈ [0, 1], respectively PΓ ∈ R+, denotes the expected reward, respectively the expected overall payment, of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, such an algorithm runs in time polynomial in the instance size and 1 ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We start by providing the general procedure underlining the approximation algorithm (see Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The algorithm implements a binary search scheme to find a value η⋆ ∈ [0, 1] such that a feasibility-version of LP (4) with the objective constrained to be at most η⋆ is “approximately” feasible, while the same problem with the objective constrained to be at most η⋆ − β is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The constant β ≥ 0 will be specified later in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Algorithm 1 requires log(β) steps and, at each step, it works by determining, for a given value η ∈ [0, 1], whether there exists an “approximately” feasible solution to the following feasibility-version of LP (4)—called F for ease of 30 ARXIV PREPRINT - FEBRUARY 1, 2023 presentation—, which is obtained by dropping the objective function from LP (4) and adding a constraint enforcing that the value of the objective is at most η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' F \uf8f1 \uf8f2 \uf8f3 � i∈N � θ∈Θ � θ−i∈ ˜Θn −i xi,θ,θ−i ≤ η Constraints (4b)—(4i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The algorithm is initialized with l = 0, h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' At each iteration of the binary search scheme, the feasibility problem F with objective ≤ η = l+h 2 is solved via an ad hoc implementation of the ellipsoid method (see the following for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' If F is found to be infeasible by the ellipsoid method, the algorithm sets l ← η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Otherwise, if F is found to be “approximately” feasible, the algorithm sets h ← η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, the procedure is repeated with the updated values of l and h, and it terminates when it determines a value η⋆ = h such that F with objective ≤ η⋆ is “approximately” feasible and F with objective ≤ η⋆ − β is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='21 In the following, we first describe in details the ad hoc implementation of the ellipsoid method employed by Al- gorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, we provide a bound on the principal’s expected utility in an optimal DISC menu of randomized contracts in terms of the value η⋆ found by Algorithm 1, as well as a η⋆-depending bound on the value of the solution returned by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Finally, we put all the bounds together in order to prove the statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Algorithm 1 Approximation algorithm introduced in the proof of Theorem 8 Input: Bayesian principal-multi-agent problem instance I := (N, Θ, Ω, A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Multiplicative approximation factor α ∈ (0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Additive approximation error ρ > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Approximate separation oracle Oα(·, ·, ·, ·) for Constraints (4e) 1: Initialization: l ← 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' h ← 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' H ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' H⋆ ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' β ← ρ 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ǫ ← ρ 4| ˜Θn| 2: while h − l > β do 3: η ← h+l 2 4: Run ad hoc ellipsoid method on F with objective ≤ η, using additive error ǫ as input in the calls to the approximate separation oracle Oα(·, ·, ·, ·) for Constraints (4e) 5: H ← {Constraints (4e) found to be violated during the ellipsoid method} 6: if ellipsoid method returned infeasible then 7: l ← η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' H⋆ ← H 8: else 9: h ← η 10: return η⋆ ← h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Optimal solution to LP (3) in which only the variables tθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a corresponding to the dual constraints in H⋆ are specified Implementation of the ellipsoid method Given a point (xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i) in the variable domain of LP (4) and η ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' our implementation of the ellipsoid method employs an ad hoc separation oracle to determine whether the point (xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ−i) is “approximately” feasible for problem F with objective ≤ η or there exists a constraint of F that is violated in such a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In the latter case, the separation oracle returns the violated constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' First, the oracle checks if one among Constraints (4b)—(4d) and Constraints (4f)—(4i) is violated, which can be done in polynomial time by checking them one by one, since such constraints are polynomially many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' If a violated constraint is found, the oracle returns it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' If all the Constraints (4b)—(4d) and the Constraints (4f)—(4i) are not violated, the oracle has to check the exponentially-many Constraints (4e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In order to do so, for every tuple of agents’ types θ ∈ ˜Θn, the oracle runs the procedure Oα(·, ·, ·, ·), feeding it with the following inputs: the instance I, the tuple θ, weights w ∈ Rnℓ such that wi,a = min{yi,θ,a, 2} for all i ∈ N and a ∈ Ai,θi, and an additive error ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' If the call Oα(I, w, θ, ǫ) returns a tuple of agents’ actions a ∈ An,θ such that λθRθ,a − � i∈N wi,ai ≤ 0, then it also holds that αλθRθ,a − � i∈N yi,θ,ai ≤ ǫ for every a ∈ An,θ, since wi,ai ≤ yi,θ,ai for all i ∈ N by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' If this happens for every tuple of agents’ types θ ∈ ˜Θn, the separation oracle then concludes that F is “approximately” feasible, meaning that Constraints (4e) are satisfied up to a reward-multiplying approximation factor α and an additive error ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Instead,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' if for some θ ∈ ˜Θn the call Oα(I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' ǫ) returns an action profile a ∈ An,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='θ such that λθRθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='a − � i∈N wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='ai > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' then it must be 21Notice that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' in the case in which the ad hoc implementation of the ellipsoid method concludes that F is “approximately” feasible at every iteration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' when it always returns infeasible,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' we can perform a similar analysis by observing that there always exists a feasible solution with objective 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' while all the feasible solutions have objective at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 31 ARXIV PREPRINT - FEBRUARY 1, 2023 wi,ai ≤ 1 for all i ∈ N (as rewards belong to [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Hence, it must be wi,ai = yi,θ,ai for all i ∈ N, and, thus, λθRθ,a − � i∈N yi,θ,ai = λθRθ,a − � i∈N wi,ai > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, the separation oracle concludes that the feasibility problem F is infeasible and outputs the constraint in Constraints (4e) related to θ ∈ ˜Θn and a ∈ An,θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Bounding the value of an optimal solution Next, prove that αROPT − POPT − |˜Θn|ǫ ≤ η⋆ for any optimal DSIC menu of randomized contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' First, let us recall that the binary search scheme in Algorithm 1 terminates with an η⋆ ∈ [0, 1] such that the ad hoc im- plementation of the ellipsoid method applied to F with objective ≤ η⋆ concludes that the problem is “approximately” feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This implies that there exists a point (xi,θ,θ−i, yi,θ,a, zi,θ,θ,a,a′, di,θ,θ−i) in the variable domain of LP (4) such that Constraints (4b)—(4d) and Constraints (4f)—(4i) are satisfied and, additionally, αλθRθ,a − � i∈N yi,θ,ai ≤ ǫ for every tuple of agents’ types θ ∈ ˜Θn and tuple of agents’ actions a ∈ An,θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In the following, we define a modified version of LP (4) (see LP (6) below) and we show that (xi,θ,θ−i, yi,θ,a, zi,θ,θ,a,a′, di,θ,θ−i) is a feasible solution to such a problem having value at most η⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' min � i∈N � θ∈Θ � θ−i∈ ˜Θn −i xi,θ,θ−i s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (6a) α λθ Rθ,a − � i∈N yi,θ,ai ≤ ǫ ∀θ ∈ ˜Θn, ∀a ∈ An,θ (6b) Constraints (4b)—(4d) and (4f)—(4i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Since Constraints (4b)—(4d) and (4f)—(4i) are satisfied by (xi,θ,θ−i, yi,θ,a, zi,θ,θ,a,a′, di,θ,θ−i), we only need to show that also Constraints (6b) are satisfied by such a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' By contradiction, suppose that Constraint (6b) relative to θ ∈ ˜Θn and a ∈ An,θ is violated by (xi,θ,θ−i, yi,θ,a, zi,θ,θ,a,a′, di,θ,θ−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, it must the case that αλθRθ,a − � i∈N yi,θ,ai − ǫ > 0, contradicting the fact the ellipsoid method classified problem F with objective ≤ η⋆ as “approximately” feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This shows that (xi,θ,θ−i, yi,θ,a, zi,θ,θ,a,a′, di,θ,θ−i) is feasible for LP (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' The dual formulation of LP (6) reads as follows: max � θ∈ ˜Θn � a∈An,θ tθ,a (αλθRθ,a − ǫ) − � i∈N � θ∈ ˜Θn λθ � a∈Ai,θi � ω∈Ω Fi,θi,a,ω yi,θ,a,ω s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (7a) � a∈An,θ:ai=a tθ,a ≤ ξi,θ,a ∀i ∈ N, ∀θ ∈ ˜Θn, ∀a ∈ Ai,θi (7b) Constraints (2b)—(2d) and (2f)—(2i), where, for ease of presentation, we used the same variable names as in LP (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' By strong duality, we have that the optimal value of LP (7) is at most η⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, since an optimal DSIC menu of randomized contracts identifies an optimal solution to LP (3) and such a solution is clearly feasible for LP (7), we have that the optimal value of LP (7) is at least α ROPT − POPT − |˜Θn|ǫ, (8) where we used the fact that, in any feasible solution, it holds � a∈An,θ tθ,a ≤ 1 for every θ ∈ ˜Θn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' This proves that αROPT − POPT − |˜Θn|ǫ ≤ η⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Bounding the value of the solution returned by Algorithm 1 Next, we show that Algorithm 1 gives as output a solution with value at least η⋆ − β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Let H⋆ ⊂ ˜Θn × An be the set of tuples of agents’ types and tuples of actions corresponding to Constraints (4e) which are identified as violated by the ad hoc ellipsoid method during the last iteration of the binary search scheme in which it returned infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' It is immediate to see that, during such an iteration, the ellipsoid method is applied to the feasibility problem F with objective ≤ l with l ≤ η⋆ − β, by definition of η⋆ and given how the binary search scheme terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' LP (4) with only the Constraints (4e) corresponding to elements in H⋆ (and all the other Constraints (4b)—(4d) and (4f)—(4i)) is infeasible, and the ellipsoid method guarantees that the elements in H⋆ are polynomially many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, the dual of such an LP is LP (3) in which only the variables tθ,a corresponding to the elements in H⋆ are 32 ARXIV PREPRINT - FEBRUARY 1, 2023 specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Formally, it can be written as: max � (θ,a)∈H⋆ λθtθ,a Rθ,a − � i∈N � θ∈ ˜Θn λθ � a∈Ai,θi � ω∈Ω Fi,θi,a,ω yi,θ,a,ω s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' (9a) � a∈An,θ:ai=a∧(θ,a)∈H tθ,a ≤ ξi,θ,a ∀i ∈ N, ∀θ ∈ ˜Θn, ∀a ∈ Ai,θi (9b) Constraints (2b)—(2d) and (2f)—(2i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' By strong duality, LP (9) has optimal value at least η⋆ − β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, an optimal solution can be computed in poly- nomial time since there are polynomially-many constraints in H⋆ and, thus, LP (9) has polynomially-many variables and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Putting all together We conclude the proof by providing the desired approximation guarantees for an optimal solu- tion to LP (9) returned by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Let APX be the value of an optimal solution to LP (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Moreover, set β := ρ 4 and ǫ := ρ 4| ˜Θn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, APX ≥ η⋆ − β ≥ αROPT − POPT − |˜Θn|ǫ − β ≥ αROPT − POPT − ρ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Finally, given a feasible solution to LP (9), by applying Lemma 9, we can recover in polynomial time a feasible solution to LP (2) with the same objective function value (notice that any solution that is feasible for LP (9) is also feasible for LP (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, by applying Theorem 7 with ǫ = ρ nτ(|i|)+1 to the just computed solution, we can recover in polynomial time a feasible solution to Problem (1), which corresponds to a DISC menu of randomized contracts with principal’s expected utility at least APX − ρ 2 − ǫ(n τ(|I|) + 1) = αROPT − POPT − ρ, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In Bayesian principal-multi-agent problem instances that (i) have succinct rewards specified by an IR- supermodular function and (ii) satisfy the FOSD condition, for any ρ > 0, the problem of computing an optimal menu of randomized contracts admits an algorithm returning a menu with principal’s expected utility at least OPT − ρ in time polynomial in the instance size and 1 ρ, where OPT is the value of the optimal principal’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We show that the problem admits a polynomial-time approximate separation oracle O1(·, ·, ·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, the result directly follows from Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' A call O1(I, w, θ, ǫ) to the approximation oracle can simply implement the polynomial-time algorithm for non- Bayesian problem (see Theorem 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Indeed, it is sufficient to rescale the function g (and hence the rewards) by a factor λθ, while replacing each value �Pi,a with the weight wi,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' It is easy to see that the arguments proving Theorem 4 continue to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In Bayesian principal-multi-agent problem instances with succinct rewards specified by a DR- submodular function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' the problem of computing an optimal menu of randomized contracts admits a polynomial-time approximation algorithm which,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' for any ǫ > 0 given as input,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' outputs a menu providing the principal with an expected utility at least of (1 − 1/e)RΓ − PΓ − ǫ for each menu of randomized contracts Γ = {γθ}θ∈Θn with high probability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' where RΓ ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' respectively PΓ ∈ R+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' denotes the expected reward,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' respectively the expected payment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' in contract p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' We show that the problem admits a polynomial-time approximate separation oracle O1−1/e(·, ·, ·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Then, the result readily follows from Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' In particular, a call O1−1/e(I, w, θ, ǫ) to the oracle oracle can be implemented by means of the polynomial-time approximation algorithm introduced for non-Bayesian instances (see Theorem 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Indeed, we can rescale the reward function g (and hence the rewards) by a factor λθ, while we can replace the value �Pi,a with the weights wi,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' It is easy to see that Theorem 6 continues to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' Finally, by Theorem 6, it is easy to show that the approximation guarantees of the oracle hold with high probability Indeed, it is sufficient to apply a union bound over the polynomially-many calls to the oracle in order to get that the approximation guarantees of all the calls hold simultaneously with high probability, proving the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'} +page_content=' 33' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFRT4oBgHgl3EQf0zhG/content/2301.13654v1.pdf'}