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+arXiv:2301.13654v1  [cs.GT]  31 Jan 2023
+MULTI-AGENT CONTRACT DESIGN: HOW TO COMMISSION
+MULTIPLE AGENTS WITH INDIVIDUAL OUTCOMES
+ARXIV PREPRINT
+Matteo Castiglioni
+Politecnico di Milano
+matteo.castiglioni@polimi.it
+Alberto Marchesi
+Politecnico di Milano
+alberto.marchesi@polimi.it
+Nicola Gatti
+Politecnico di Milano
+nicola.gatti@polimi.it
+February 1, 2023
+ABSTRACT
+We study hidden-action principal-agent problems with multiple agents. 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. Previous
+works on multi-agent problems study models where the principal observes a single outcome deter-
+mined by the actions of all the agents. 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. 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. This considerably enhances the principal’s contracting capabilities, by allowing
+them to pay each agent on the basis of their individual result. 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. 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. These two properties naturally model two common real-world phenomena, namely disec-
+onomies and economies of scale. 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. 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. Such a reduction is needed to prove our main
+positive results in the rest of the paper. 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. 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. In conclusion, we extend our
+positive results to Bayesian instances. 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.
+This is non-trivial, since in general the problem may not admit a maximum, but only a supremum.
+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. We prove that such an oracle can be implemented in polynomial time by exploiting our posi-
+tive results on non-Bayesian instances. Surprisingly, this allows us to (almost) match the guarantees
+obtained for non-Bayesian instances.
+
+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. 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. 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. The principal gets a reward associated with the realized outcome, while an agent incurs in a cost when
+performing an action. Thus, the principal’s goal is to incentivize agents to undertake actions which result in profitable
+outcomes. 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.
+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. 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). 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.
+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.g., crowdsourcing platforms (Ho et al., 2016), blockchain-based smart con-
+tracts (Cong and He, 2019), and healthcare (Bastani et al., 2016).
+The computational aspects of principal-agent problems with a single agent have been widely investigated in the
+literature. Instead, only few works study problems with multiple agents. Some notable examples are the papers
+by Babaioff et al. (2006) and Emek and Feldman (2012), and the very recent preprint by Duetting et al. (2022). These
+works address models where the principal observes a single outcome determined by the actions of all the agents. 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.
+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. Our model fits many real-world applications. 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. 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., 2006; Emek and Feldman, 2012; Duetting et al., 2022). In-
+deed, as we discuss later in Section 1.2, such issues originate from the appearance of externalities among the agents,
+which are instead not present in our setting.
+1.1
+Original Contributions
+We investigate the computational complexity of finding optimal contracts in our principal-multi-agent problems with
+agents’ individual outcomes. Our analysis revolves around two properties of principal’s rewards, which we call IR-
+supermodularity and DR-submodularity. 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. These two properties naturally model two common real-world phenomena,
+namely diseconomies and economies of scale, respectively.
+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.e., their action costs and the probability distributions that
+their actions induce over (individual) outcomes. 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.
+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. 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. 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. At the end of the section (more precisely in Section 3.3), 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. This will be useful
+to derive our positive result in the following Section 4, and it may also be of independent interest.
+2
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+In Section 4, we provide our main results on non-Bayesian instances with IR-supermodular principal’s rewards. 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. Then, we show how to circumvent such a negative result by introducing a mild regularity
+assumption. 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. 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.
+In Section 5, we switch our attention to non-Bayesian instances with DR-submodular principal’s rewards. 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. Next,
+we complement such a negative result by providing a polynomial-time approximation algorithm for the problem. In
+particular, we exploit the reduction to matroid optimization introduced in Section 3 and a result by Sviridenko et al.
+(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.
+Finally, we conclude the paper, in Section 6, by providing our results on Bayesian principal-multi-agent problems.
+First, we extend the model recently introduced by Castiglioni et al. (2022b) to our multi-agent setting. 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. In such a setting, the principal is better off committing to a
+menu of randomized contracts rather than a single contract. 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. 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. 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. Notice that this step is non-trivial, since in
+general the principal’s optimization problem may not admit a maximum, but only a supremum. 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. Such an oracle can be implemented
+by using the algorithms developed in Sections 4 and 5 for non-Bayesian instances.
+1.2
+Related Works
+Next, we survey the most-related computational works on hidden-action principal-agent problems.
+Works on Principal-Agent Problems with a Single Agent.
+Most of these works focus on non-Bayesian settings.
+Among the most related to ours, Dutting et al. (2021) and D¨utting et al. (2022) study models whose underlying struc-
+ture is combinatorial. 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). 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. (2019) use the computational lens to analyze the efficiency
+(in terms of principal’s expected utility) of linear contracts with respect to optimal ones. Recently, some works also
+considered the more realistic Bayesian settings (Guruganesh et al., 2021; Alon et al., 2021; Castiglioni et al., 2022a,c).
+In particular, Castiglioni et al. (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.e., non-randomized) contracts,
+which were previously studied in (Guruganesh et al., 2021; Alon et al., 2021).
+Works on Principal-Agent Problems with Multiple Agents.
+All the previous works on multi-agent settings are
+limited to non-Bayesian instances. Babaioff et al. (2006) are the first to study a model with multiple agents (see also
+its extended version (Babaioff et al., 2012) and its follow-ups (Babaioff et al., 2009, 2010)). 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. 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. Babaioff et al. (2006) show that finding an optimal contract is #P-complete even when the outcome-
+determining function is represented as a “simple” read-once network. Emek and Feldman (2012) extend the work
+by Babaioff et al. (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. Finally, a very recent preprint by Duetting et al. (2022) considerably extends previous
+works by providing constant-factor approximation algorithms for problems with submodular and XOS rewards.
+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; 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.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.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. 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. 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. Each tuple ω ∈ Ωn has an associated reward to the principal, which we
+denote by rω ∈ [0, 1]. 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.
+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. This is in contrast with previous works
+on principal-multi-agent problems (see, e.g., (Babaioff et al., 2006; Emek and Feldman, 2012; Duetting et al., 2022)),
+which assume that the principal can only observe a single outcome that is jointly determined by the tuple of all the
+agents’ actions.
+2.1
+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. 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.5 Notice that the assumption that payments are non-negative (i.e., they
+can only be from the principal to agents) is common in contract theory, where it is known as limited liability (Carroll,
+2015). 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.
+Given a contract p ∈ Rn×m
++
+, each agent i ∈ N selects an action such that:
+1. it is incentive compatible (IC), i.e., it maximizes their expected utility among actions in A;
+2. it is individually rational (IR), i.e., 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).
+For ease of presentation, we make the following w.l.o.g. assumption:
+Assumption 1 (Null action). There exists an action a∅ ∈ A such that ci,a∅ = 0 for all i ∈ N.
+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.l.o.g. on incentive compatibility only. 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. Formally, it holds A∗
+i (p) := arg maxa∈A {Pi,a − ci,a}. 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.
+2For ease of presentation, we assume that all the agents share the same action set and outcome set. 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.
+3In this paper, given a finite set X, we denote by ∆X the set of all the probability distributions defined over elements of X.
+4For ease of presentation, costs and rewards are in [0, 1]. All the results can be easily generalized to an arbitrary range.
+5W.l.o.g., 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. 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. 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.
+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; formally,
+Pi,a :=
+�
+p ∈ Rn×m
++
+| a ∈ A∗
+i (p)
+�
+.
+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. Thus, it is necessary to adopt a suitable tie-breaking-rule assumption.
+Remark 1 (On classical tie-breaking rules). 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.g., (Dutting et al.,
+2021)). Such an assumption is unreasonable in our setting. 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.
+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. 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. Given that
+the actions in a∗ are IC under p, we assume w.l.o.g. that the agents stick to such recommendations.
+In conclusion, the principal’s optimization problem reads as follows:
+Definition 1 (Principal’s Optimization Problem). 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.
+2.2
+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. Thus, we work with a succinct representation of
+principal’s rewards, which we formally introduce in the following. We remark that, with arbitrary rewards, an optimal
+contract can be found in time polynomial in the instance size (i.e., in time depending polynomially on mn), as we
+show in Section 3.2.
+We say that a principal-multi-agent problem instance (N, A, Ω) has succinct rewards if:
+1. 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
++;
+2. 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.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].7 Moreover, in the following, we assume that g : Rnq
++ → R can be accessed through an
+oracle that, given ω ∈ Ωn, outputs g(ω).8
+In this work, we make the following common assumption on principal’s rewards:
+Assumption 2 (Increasing rewards). The principal’s reward function g : Rnq
++ → R is increasing; formally, it holds
+that g(ω) ≥ g(ω′) for all ω, ω′ ∈ Rnq
++ : ω ≥ ω′.
+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.
+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.
+7In this paper, given a positive natural number x ∈ N>0, we let [x] := {1, . . . , x} be the set of the first x natural numbers.
+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. 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). All the results in the paper can be easily extended
+to also account for this additional (arbitrarily small) approximation.
+5
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+Definition 2 (DR-submodularity and IR-supermodularity). 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′).
+Moreover, a reward function g : Rnq
++ → R is increasing-return supermodular (IR-supermodular)if its opposite function
+−g is DR-submodular.
+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. Indeed, as shown by Bian et al. (2017), when g is
+continuously differentiable, g is DR-submodular if and only if:
+∇g(ω) ≥ ∇g(ω′)
+∀ω, ω′ ∈ Rnq
++ : ω ≥ ω′.
+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. This property is satisfied in many real-world scenarios,
+as we show in the following specific example.
+Example 1 (Selling multiple products). 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). 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. 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. 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. This may be due to the fact that, e.g., the firm has to
+sustain much higher operational costs in order increase its selling capacity. 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.g., the fixed costs are more efficiently covered).
+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. First, in Section 3.1, we introduce
+some preliminary definitions on matroids and optimization problems over matroids. Then, in Section 3.2, we provide
+the reduction.
+We conclude the section with Section 3.3, 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.
+This result will be useful in the following Section 4.
+3.1
+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. the empty set is independent, i.e., ∅ ∈ I;
+2. every subset of an independent set is independent, i.e., for S′ ⊆ S ⊆ G, if S ∈ I then S′ ∈ I;
+3. if S ∈ I and S′ ∈ I are two independent sets such that S has more elements than S′, i.e., |S| > |S′|, then
+there exists an element x ∈ S \ S′ such that S′ ∪ {x} ∈ I.
+Any subset S ⊆ G such that S /∈ I is said to be dependent. 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. We denote by B(M) ⊂ 2G the set of the bases of M. We refer the reader to (Schrijver et al., 2003)
+for a detailed treatment of matroids.
+In the following, we will also consider optimization problems defined over matroids. 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).
+3.2
+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). A matroid M := (G, I) is a 1-partition matroid if there exists d ∈ N+ subsets
+Gi ⊆ G of ground elements such that:
+1. G = �
+i∈[d] Gi and Gi ∩ Gj = ∅ for all i, j ∈ [d] : i ̸= j;
+2. I = {S ⊆ G : |S ∩ Gi| ≤ 1 ∀i ∈ [d]}.
+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. 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]. 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.
+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). 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. GI
+i := {(i, a) : a ∈ A} for all i ∈ N, with GI := �
+i∈[d] GI
+i ;
+2. II :=
+�
+S ⊆ GI : |S ∩ GI
+i | ≤ 1 ∀i ∈ [d]
+�
+.
+It is immediate to check that MI is indeed a 1-partition matroid. Moreover, its bases correspond one-to-one to
+agents’ action profiles a ∈ An. In particular, an independent set S ∈ II of MI assigns an action to each agent
+in NS :=
+�
+i ∈ N : |S ∩ GI
+i | = 1
+�
+; we denote by aS,i ∈ A the action associated to agent i ∈ NS. 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. 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; formally, aS,i = a∅ for all i ∈ N \ NS.
+The following theorem formalizes our reduction:
+Theorem 1. 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.
+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. 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.
+Lemma 1. There always exists a base S∗ ∈ B(MI) of MI such that f I(S∗) = maxS∈II f I(S).
+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. 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. 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.
+3.3
+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. In order to do this, we first need some additional notation. 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). 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.e., 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′). Analogously, we define S ∨ S′ as the partition-wise “minimum” of the two sets.
+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. Formally:
+Definition 5 (Ordered-supermodular function). 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′).
+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. 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.
+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; Bach, 2019).9
+Theorem 2. 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.
+Corollary 1. The problem of maximizing an ordered-supermodular function over a 1-partition matroid admits a
+polynomial-time algorithm.
+4
+Principal-Multi-Agent Problems with IR-supermodular Rewards
+In this section, we study principal-multi-agentproblems with succinct rewards specified by IR-supermodularfunctions.
+First, in Section 4.1, we prove that in such setting the problem of computing an optimal contract is inapproximable in
+polynomial time. Then, in Section 4.2 we show that, under mild assumptions, the problem can be solved in polynomial
+time.
+4.1
+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. 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. Such a problem is known to be NP-hard (Raz, 1998; Arora et al., 1998). We
+refer the reader to Appendix B for a formal definition of the problem.
+Our inapproximability result formally reads as follows:
+Theorem 3. 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.
+Indeed, the proof of Theorem 3 provides an even stronger hardness result. 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
++
+.
+Formally, the following corollary holds:
+Corollary 2. 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.
+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.
+9We recall that a ring of sets is a family of sets R that is closed under both union and intersection. Formally, given any two sets
+S, S′ ∈ R, it holds S ∪ S′ ∈ R and S ∩ S′ ∈ R (Birkhoff, 1937).
+8
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+4.2
+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. 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.
+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.g., (Tadelis and Segal, 2005)).
+Moreover, such a condition is reasonably satisfied in many real-world settings. 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. 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. Naturally, the salesperson undertaking an higher level of effort in selling products will result in a bigger
+probability of generating large volumes of sales.
+In order to formally define the FOSD condition, we first need to introduce some additional notation. 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′.
+Moreover, for ease of presentation, with a slight abuse of notation and w.l.o.g. we assume that the actions of each
+agent i ∈ N are re-labeled so that A = {a1, . . . , aℓ} with ci,aj ≤ ci,aj+1 for every j ∈ [ℓ − 1]. Then, we have the
+following definition:
+Definition 6 (First order stochastic dominance). 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.
+Remark 2. A condition similar to Definition 6, called monotone likelihood ratio property (MLRP), has been consid-
+ered by D¨utting et al. (2019) limited to the case in which outcomes are identified by scalar values. In such settings, the
+MLRP is strictly stronger than the FOSD condition. Our definition of FOSD generalizes the classical FOSD condition
+(see, e.g., (Tadelis and Segal, 2005)) from settings in which the outcomes are scalar values to those where they are
+vector values.
+Next, we prove how to design a polynomial-time algorithm for the problem of finding an optimal contract by exploiting
+the FOSD condition. Intuitively, the idea of the proof is the following. 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). Moreover, by Corollary 1, if f I is ordered-supermodular we can solve the optimization problem
+in polynomial time. 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.
+First, we prove the following preliminary result which follows from (Østerdal, 2010).
+Lemma 2. 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.
+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.
+Lemma 3. 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.
+Finally, Lemma 3 allows us to prove the main positive result of this section:
+Theorem 4. 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.
+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. First, similarly to the case of IR-supermodular reward functions, we provide a strong negative
+result. 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.
+In order to prove the negative result, we provide a reduction from the promise version of the well-known
+INDEPENDENT-SET problem. 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. Such a problem is known to be NP-
+hard for any α > 0 (H˚astad, 1999; Zuckerman, 2007). This is exploited by our reduction in order to prove Theorem 5.
+The reader can find more details on the definition of the promise version of INDEPENDENT-SET in Appendix D.
+Theorem 5. 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.
+Next, we complement the inapproximability result in Theorem 5 by providing a polynomial-time approximation al-
+gorithm for the problem. 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. 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.
+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). To
+do this, we first need to introduce some additional notation.
+For ease of presentation, in the rest of this section we will make the following w.l.o.g. assumption:
+Assumption 3 (Null outcome). 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∅}.
+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}. 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. We denote such a set by ˜Ω ⊆ Rq
++, and let Fi,S,ω be the probability that Fi,S assigns to ω ∈ ˜Ω. Moreover,
+we let ˜Ωn :=×i∈N ˜Ω. Finally, we overload notation and let RS := �
+ω∈˜Ωn rω
+�
+i∈N Fi,S,ωi for any S ⊆ GI. 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).
+We are now ready to provide the formal definition of the extension of f I:
+Definition 7 (Extension of f I). 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.
+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. Formally:
+Lemma 4. 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.
+Lemma 4 allows us to apply a result by Sviridenko et al. (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. This immediately gives the following result:
+10
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+Theorem 6. 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∗).
+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. In particular, we extend the Bayesian model recently introduced
+by Castiglioni et al. (2022b) to multi-agent settings.
+First, in Section 6.1 we formally introduce Bayesian principal-multi-agent problems and all their related concepts.
+Then, Section 6.2 provides a formulation of the computational problem that the principal has to solve in Bayesian
+settings. Next, in Section 6.3 we show how such a problem can be “approximately formulated” as an LP with
+exponentially-many variables and polynomially-many constraints. Finally, in Section 6.4 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.
+6.1
+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.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. 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.11 Action costs and distributions over outcomes are extended so that they also
+depend on the agent’s type; formally, they are denoted as Fi,θ,a and ci,θ,a, where θ ∈ Θ is the type of agent i ∈ N.
+Similarly, we extend the definition of expected reward, denoted as Rθ,a. Moreover, w.l.o.g., we modify Assumption 1
+so that the null action a∅ now satisfies ci,θ,a∅ = 0 for all i ∈ N and θ ∈ Θ. 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.
+Following the line of Castiglioni et al. (2022b), we consider the case in which the principal commits to a menu of
+randomized contracts. In our multi-agent setting, a randomized contract is defined as a probability distribution γ
+supported on Rn×m
++
+. 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.
+The interaction between the principal and agents having types specified by θ ∼ λ goes as follows:
+1. the principal commits to a menu of randomized contracts Γ = (γθ)θ∈Θn;
+2. each agent i ∈ N reports a type ˆθi ∈ Θ to the principal (possibly different from their type θi);
+3. 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;
+4. 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).
+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. Thus, in the rest of this section, whenever we refer to a contract
+10For ease of exposition, all agents share the same set Θ. Our results can be easily extended to the case of agent-specific sets.
+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. Hence, the running time of any polynomial-time algorithm for such a problem
+must depend polynomially on the size of supp(λ). 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.
+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.
+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).
+As in single-agent settings (Castiglioni et al., 2022b), it is possible to focus w.l.o.g. on menus of randomized contracts
+that are dominant-strategy incentive compatible (DSIC).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).
+6.2
+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. The formulation that we propose in the following is specifically tailored so as to ease
+the design of our approximation algorithm.
+As a first step, we show that we can focus w.l.o.g. on randomized contracts γθ having a finite support supp(γθ) ⊆
+Rn×m
++
+. Such a result is already known for single-agent settings (see Lemma 1 in (Castiglioni et al., 2022b) and
+Theorem 1 in (Gan et al., 2022)), but it can be easily generalized to our multi-agent problems. In particular, since
+in our model there are no externalities among the agents, it is immediate to adapt the results of Castiglioni et al.
+(2022b) and Gan et al. (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. Moreover, such agent i’s payment scheme is
+such that action a is IC when the type of agent i is θi. Formally:
+Lemma 5. 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.e.,
+the agent i’s payment scheme under contract p).
+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. 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;
+• 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;
+• 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.
+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. Before doing that, for ease of presentation, we introduce some
+additional notation. 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. Moreover, given any θ ∈ Θn, we let θ−i be the
+tuple obtained by dropping agent i’s type θi from θ. 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). Finally, for every agent i ∈ N, we denote by
+12It is easy to show that focusing on DSIC menus of randomized contracts is w.l.o.g. by using a revelation-principle-style
+argument. See the book by Shoham and Leyton-Brown (2008) for some examples of these kinds of arguments.
+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.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.t.
+(1a)
+�
+a∈A
+ξi,θ,a
+��
+ω∈Ω
+Fi,θi,a,ω pi,θ,a,ω − ci,θi,a
+�
+≥
+�
+a∈A
+ξi,(θ,θ−i),a max
+a′∈A
+��
+ω∈Ω
+Fi,θi,a′,ω pi,(θ,θ−i),a,ω − ci,θi,a′
+�
+∀i ∈ N, ∀θ ∈ ˜Θn, ∀θ ∈ Θ
+(1b)
+�
+a∈A
+ξi,(θ,θ−i),a = 1
+∀i ∈ N, ∀θ ∈ Θ, ∀θ−i ∈ ˜Θn
+−i
+(1c)
+�
+a∈An:ai=a
+tθ,a = ξi,θ,a
+∀i ∈ N, ∀θ ∈ ˜Θn, ∀a ∈ A
+(1d)
+tθ,a ≥ 0
+∀θ ∈ ˜Θn
+−i, ∀a ∈ An
+(1e)
+ξi,(θ,θ−i),a ≥ 0
+∀i ∈ N, ∀θ ∈ Θ, ∀θ−i ∈ ˜Θn
+−i, ∀a ∈ A
+(1f)
+pi,(θ,θ−i),a,ω ≥ 0
+∀i ∈ N, ∀θ ∈ Θ, ∀θ−i ∈ ˜Θn
+−i, ∀a ∈ A, ∀ω ∈ Ω,
+(1g)
+where Objective (1a) is the principal’s expected utility for the menu of randomized contracts encoded by the variables
+in the problem, Constraints (1b) specify the conditions ensuring that the menu is DSIC (for θ ̸= θi), as well as the
+conditions guaranteeing that any action a such that ξi,θ,a > 0 is IC for an agent i of type θi under the payments defined
+by variables pi,θ,a,ω, while Constraints (1c) and (1d) ensure that the menu of randomized contracts is well defined.
+Notice that Problem (1) is defined in terms of sup rather than max. This is because, as shown in (Castiglioni et al.,
+2022b), even in single-agent settings the problem of computing an optimal DSIC menu of randomized contracts may
+not admit a maximum. In the following, for ease of presentation, we let SUP be the optimal value of Problem (1) (i.e.,
+the value of the supremum).
+6.3
+An “Approximately-optimal” LP Formulation
+As a preliminary step towards the design of our approximation algorithm (see Section 6.4), 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.
+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. This is the set of all actions that are IC for an agent i of type θ under at least one contract; formally,
+Ai,θ :=
+�
+a ∈ A | ∃p ∈ Rn×m
++
+: a ∈ A∗
+i,θ(p)
+�
+.
+First, we can prove the following useful result:
+Lemma 6. 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.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. 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).
+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). 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.
+14In the rest of the section, we always assume that the size of a problem instance is expressed in terms of number of bits.
+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). 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.t.
+(2a)
+�
+a∈Ai,θi
+��
+ω∈Ω
+yi,θ,a,ω Fi,θi,a,ω − ξi,θ,a ci,θi,a
+�
+≥
+�
+a∈Ai,θ
+γi,θ,θ,a ∀i ∈ N, ∀θ ∈ ˜Θn, ∀θ ∈ Θ
+(2b)
+γi,θ,θ,a ≥
+�
+ω∈Ω
+yi,(θ,θ−i),a,ω Fi,θi,a′,ω − ξi,(θ,θ−i),a ci,θi,a′
+∀i ∈ N, ∀θ ∈ ˜Θn, ∀θ ∈ Θ, ∀a ∈ Ai,θi, ∀a′ ∈ Ai,θi
+(2c)
+�
+a∈Ai,θ
+ξi,(θ,θ−i),a = 1
+∀i ∈ N, ∀θ ∈ Θ, ∀θ−i ∈ ˜Θn
+−i
+(2d)
+�
+a∈An,θ:ai=a
+tθ,a = ξi,θ,a
+∀i ∈ N, ∀θ ∈ ˜Θn, ∀a ∈ Ai,θi
+(2e)
+tθ,a ≥ 0
+∀θ ∈ ˜Θn, ∀a ∈ An,θ
+(2f)
+ξi,(θ,θ−i),a ≥ 0
+∀i ∈ N, ∀θ ∈ Θ, ∀θ−i ∈ ˜Θn
+−i, ∀a ∈ Ai,θ
+(2g)
+yi,(θ,θ−i),a,ω ≥ 0
+∀i ∈ N, ∀θ ∈ Θ, ∀θ−i ∈ ˜Θn
+−i, ∀a ∈ Ai,θ, ∀ω ∈ Ω
+(2h)
+γi,θ,θ,,a
+∀i ∈ N, ∀θ ∈ ˜Θn, ∀θ ∈ Θ, ∀a ∈ Ai,θ.
+(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).
+Lemma 7. For every instance of Bayesian principal-multi-agent problem, it holds LP ≥ SUP.
+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. However, the converse is not true in general, i.e., 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. Thus, it might be the case that SUP < LP. This is caused by the existence of what we call irregular feasible
+solutions to LP (2):
+Definition 8. 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. A feasible
+solution to LP (2) is said to be regular if it is not irregular.
+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. However, the same is not true for irregular solutions, as the operation above is clearly ill defined in that
+case. 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. Formally:
+Lemma 8. 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.
+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). Formally:
+Theorem 7. 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.
+14
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+6.4
+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., 2012).
+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. This, together with some ad
+hoc modifications to the ellipsoid method, allows us to design the desired approximation algorithm for the problem
+of interest. 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. Instead, in instances having DR-submodular succinct
+rewards, this is not possible, and, thus, we need an approximate separation oracle.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). 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. 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.t.
+(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).
+Lemma 9. For every instance of Bayesian principal-multi-agent problem, LP (2) and LP (3) have the same optimal
+value. 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.
+By Lemma 9, we can solve LP (3) instead of LP (2). The dual problem of LP (3) reads as follows:16
+min
+�
+i∈N
+�
+θ∈Θ
+�
+θ−i∈ ˜Θn
+−i
+xi,θ,θ−i
+s.t.
+(4a)
+−1
+�
+(θ, θ−i) ∈ ˜Θn� �
+θ′∈Θ
+yi,(θ,θ−i),θ′ ci,θ,a +
+�
+θ′∈Θ:
+(θ′,θ−i)∈ ˜Θn
+�
+a′∈Ai,θ′
+ci,θ′,a′ zi,(θ′,θ−i),θ,a,a′
++di,θ,θ−i − 1
+�
+(θ, θ−i) ∈ ˜Θn�
+xi,θ,θ−i ≥ 0
+∀i ∈ N, ∀θ ∈ Θ, ∀θ−i ∈ ˜Θn
+−i, ∀a ∈ Ai,θ
+(4b)
+−yi,θ,θ +
+�
+a′∈Ai,θi
+zi,θ,θ,a,a′ ≥ 0
+∀i ∈ N, ∀θ ∈ ˜Θn, ∀θ ∈ Θ, ∀a ∈ Ai,θ
+(4c)
+1
+�
+(θ, θ−i) ∈ ˜Θn� �
+θ′∈��
+yi,(θ,θ−i),θ′ Fi,θ,a,ω −
+�
+θ′∈Θ:
+(θ′,θ−i)∈ ˜Θn
+�
+a′∈Ai,θ′
+Fi,θ′,a′,ω zi,(θ′,θ−i),θ,a,a′ ≥
+−1
+�
+(θ, θ−i) ∈ ˜Θn�
+λ(θ,θ−i) Fi,θ,a,ω
+∀i ∈ N, ∀θ ∈ Θ, ∀θ−i ∈ ˜Θn
+−i, ∀a ∈ Ai,θ, ∀ω ∈ Ω
+(4d)
+�
+i∈N
+yi,θ,ai ≥ λθ Rθ,a
+∀θ ∈ ˜Θn, ∀a ∈ An,θ
+(4e)
+xi,θ,θ−i
+∀i ∈ N, ∀θ ∈ θ, ∀θ−i ∈ ˜Θn
+−i
+(4f)
+yi,θ,a ≥ 0
+∀i ∈ N, ∀θ ∈ ˜Θn, ∀a ∈ Ai,θi
+(4g)
+zi,θ,θ,a,a′ ≤ 0
+∀i ∈ N, ∀θ ∈ ˜Θn, ∀θ ∈ Θ, ∀a ∈ Ai,θi, ∀a′ ∈ Ai,θi
+(4h)
+di,θ,θ−i
+∀i ∈ N, ∀θ ∈ Θ, ∀θ−i ∈ ˜Θn
+−i,
+(4i)
+where xi,θ,θ−i are dual variables that correspond to Constraints (2d), yi,θ,a to Constraints (2e), zi,θ,θ,a,a′ to Con-
+straints (2c), while di,θ,θ−i to Constraints (2b).
+15Notice that the existence of an exact oracle for instances with DR-submodular rewards would contradict Theorem 5.
+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.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. 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. As we show next, only an “approximate version” of such a
+separation oracle can be implemented in polynomial time, according to the following definition.
+Definition 9 (Approximate separation oracle). Given any α ∈ (0, 1], an approximate separation oracle for Con-
+straints (4e) is a procedure Oα(·, ·, ·, ·) which, given in input an instance I := (N, Θ, Ω, A) of Bayesian principal-
+multi-agent problem, a tuple of agents’ types θ ∈ ˜Θn, a vector w ∈ Rnℓ of weights—with wi,a denoting the vector
+component corresponding to agent i ∈ N and action a ∈ Ai,θi—, and an additive error ǫ > 0, returns a tuple of
+agents’ actions a ∈ An,θ such that:
+λθ Rθ,a −
+�
+i∈N
+wi,ai ≥ α λθ Rθ,a′ −
+�
+i∈N
+wi,a′
+i − ǫ
+∀a′ ∈ An,θ,
+in time polynomial in |I|, maxi∈N,a∈A |wi,a|, and 1
+ǫ , where |I| denotes the size of instance I.
+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.
+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). 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.
+Theorem 8. 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. Moreover, such an algorithm runs in time polynomial in the instance size and 1
+ρ.
+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. This is the last step needed to fully specify
+the approximation algorithm introduced in Theorem 8.
+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 = 1.18
+Instead, in instances with DR-submodular succinct rewards, we sow to implement an oracle Oα(·, ·, ·, ·) with
+α = 1−1/e. These implementations work by solving suitably-defined problems that resemble non-Bayesian principal-
+multi-agent instances. 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. Formally, we get the following two results:
+Corollary 3. 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.
+Corollary 4. In Bayesian principal-multi-agent problem instances with succinct rewards specified by a DR-
+submodular function, the problem of computing an optimal menu of randomized contracts admits a polynomial-time
+approximation algorithm which, for any ǫ > 0 given as input, 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,
+where RΓ ∈ [0, 1], respectively PΓ ∈ R+, denotes the expected reward, respectively the expected payment, in contract
+p.
+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.
+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). We decided to use an approximate separation oracle anyway, for ease of exposition. 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.
+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).
+17
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
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+Artificial Intelligence 307 (2022).
+Matteo Castiglioni, Alberto Marchesi, and Nicola Gatti. 2022b. Designing Menus of Contracts Efficiently: The Power
+of Randomization. https://doi.org/10.48550/ARXIV.2202.10966
+Matteo Castiglioni, Alberto Marchesi, and Nicola Gatti. 2022c. Designing Menus of Contracts Efficiently: The Power
+of Randomization. In EC ’22: The 23rd ACM Conference on Economics and Computation. 705–735.
+Lin William Cong and Zhiguo He. 2019. Blockchain disruption and smart contracts. The Review of Financial Studies
+32, 5 (2019), 1754–1797.
+Paul Duetting, Tomer Ezra, Michal Feldman, and Thomas Kesselheim. 2022. Multi-Agent Contracts. arXiv preprint
+arXiv:2211.05434 (2022).
+Paul D¨utting, Tomer Ezra, Michal Feldman, and Thomas Kesselheim. 2022. Combinatorial contracts. In 2021 IEEE
+62nd Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 815–826.
+Paul D¨utting, Tim Roughgarden, and Inbal Talgam-Cohen. 2019. Simple versus optimal contracts. In Proceedings of
+the 2019 ACM Conference on Economics and Computation. 369–387.
+Paul Dutting, Tim Roughgarden, and Inbal Talgam-Cohen. 2021. The complexity of contracts. SIAM J. Comput. 50,
+1 (2021), 211–254.
+Yuval Emek and Michal Feldman. 2012. Computing optimal contracts in combinatorial agencies. Theoretical Com-
+puter Science 452 (2012), 56–74.
+Jiarui Gan, Minbiao Han, Jibang Wu, and Haifeng Xu. 2022. Optimal Coordination in Generalized Principal-Agent
+Problems: A Revisit and Extensions. arXiv preprint arXiv:2209.01146 (2022).
+Martin Gr¨otschel, L´aszl´o Lov´asz, and Alexander Schrijver. 2012. Geometric algorithms and combinatorial optimiza-
+tion. Vol. 2. Springer Science & Business Media.
+Guru Guruganesh, Jon Schneider, and Joshua R Wang. 2021. Contracts under moral hazard and adverse selection. In
+EC ’21: The 22nd ACM Conference on Economics and Computation. 563–582.
+18
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+Johan H˚astad. 1999. Clique is hard to approximate within n1−ǫ. Acta Mathematica 182, 1 (1999), 105–142.
+Chien-Ju Ho, Aleksandrs Slivkins, and Jennifer Wortman Vaughan. 2016. Adaptive contract design for crowdsourcing
+markets: Bandit algorithms for repeated principal-agent problems. Journal of Artificial Intelligence Research 55
+(2016), 317–359.
+Ran Raz. 1998. A parallel repetition theorem. SIAM J. Comput. 27, 3 (1998), 763–803.
+Alexander Schrijver. 2000. A combinatorial algorithm minimizing submodular functions in strongly polynomial time.
+Journal of Combinatorial Theory, Series B 80, 2 (2000), 346–355.
+Alexander Schrijver et al. 2003. Combinatorial optimization: polyhedra and efficiency. Vol. 24. Springer.
+Yoav Shoham and Kevin Leyton-Brown. 2008. Multiagent systems: Algorithmic, game-theoretic, and logical founda-
+tions. Cambridge University Press.
+Maxim Sviridenko, Jan Vondr´ak, and Justin Ward. 2017. Optimal approximation for submodular and supermodular
+optimization with bounded curvature. Mathematics of Operations Research 42, 4 (2017), 1197–1218.
+Steve Tadelis and Ilya Segal. 2005. Lectures in contract theory. Lecture notes for UC Berkeley and Stanford University
+(2005).
+David Zuckerman. 2007. Linear Degree Extractors and the Inapproximability of Max Clique and Chromatic Number.
+Theory of Computing 3, 6 (2007), 103–128.
+Lars Peter Østerdal. 2010.
+The mass transfer approach to multivariate discrete first order stochastic dom-
+inance:
+Direct proof and implications.
+Journal of Mathematical Economics 46, 6 (2010), 1222–1228.
+https://doi.org/10.1016/j.jmateco.2010.08.018 The Conferences at Barcelona, Milan, New Haven, San
+Diego and Tokyo.
+19
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+A
+Proofs Omitted from Section 3
+Lemma 1. There always exists a base S∗ ∈ B(MI) of MI such that f I(S∗) = maxS∈II f I(S).
+Proof. Let S ∈ II be any independent set of MI. Clearly, by adding to S all the ground elements (i, a∅) ∈ GI
+i for
+i ∈ N \ NS, we obtain a base S′ ∈ II. 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′. This concludes the proof.
+Theorem 1. 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.
+Proof. 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). This, together with Lemma 1, proves the result.
+From (p, a) to a base.
+Let the base S ∈ II be defined so that S := {(i, ai) : i ∈ N}. 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).
+From a base to (p, a).
+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). 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).
+Theorem 2. 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.
+Proof. 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. 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. 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]. Then, it is sufficient to show that ˜f is supermodular. 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.
+Corollary 1. The problem of maximizing an ordered-supermodular function over a 1-partition matroid admits a
+polynomial-time algorithm.
+Proof. The problem can be reduced in polynomial time to the maximization of a supermodular function defined over
+a ring of sets by Theorem 2. Such a problem is known to be solvable in polynomial time; see, e.g., (Schrijver, 2000;
+Bach, 2019).
+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.
+Definition 10 (LABEL-COVER instance). 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;
+• Σ is a finite set of labels; and
+• Π := {Πe : Σ → Σ | e ∈ E} is a finite set of edge constraints.
+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. Formally, a labeling π satisfies the constraint for an edge e = (u, v) ∈ E if it
+holds that π(v) = Πe(π(u)).
+The classical LABEL-COVER problem is the search problem of finding a valid labeling for a LABEL-COVER in-
+stance given as input. In the following, we consider a different version of the problem, which is the promise problem
+associated with LABEL-COVER instances.
+Definition 11 (GAP-LABEL-COVERc,s). For any pair of numbers 0 ≤ s ≤ c ≤ 1, we define GAP-LABEL-
+COVERc,s as the following promise problem.
+• 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;
+– any labeling π satisfies less than a fraction s of the edge constraints in Π.
+• Output: Determine which of the above two cases hold.
+To prove Theorem 3, we use the following result due to Raz (1998) and Arora et al. (1998).
+Theorem 9 (Raz (1998); Arora et al. (1998)). 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.
+Now, we are ready to prove Theorem 3.
+Proof of Theorem 3. Given an approximation factor ρ > 0, we reduce from the problem GAP-LABEL-COVER1,ρ.
+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. 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. Since ρ > 0 can be an arbitrarily small constant, this is
+sufficient to prove the statement.
+Construction.
+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. The set of agents includes an agent nv for every node v ∈ U ∪ V of
+G. The outcome space has kρ dimensions, i.e., Ω = Rkρ
++ . Each agent nv, v ∈ V ∪ U has an action aσ for each label
+σ ∈ Σ. 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, . . ., kρ}. For each agent nv and each action aσ, with σ ∈ Σ, cost cn,aσ = 0 and
+aσ induces the outcome ωσ deterministically, i.e., Fn,aσ,ωσ = 1. Finally, the principal’s reward function g is defined
+as follows. For each vector ω ∈ Ωn,
+g(ω) =
+�
+(v,u)∈E
+�
+σ∈Σ
+1{ωnv,σ = 1 ∧ ωnu,Πe(σ) = 1}/|E|.
+It is easy to see that the function is IR-supermodular in [0, 1]n|Σ| and hence for all the inducible outcomes ω.19
+19It is easy to construct an arbitrary good approximation of g(·) that is IR-supermodular on all the domain Rn|Σ|
++
+. For instance,
+we can set g(ω) = eM(ωnv,σ+ωnu,Πe(σ)−2)/|E| for an arbitrary large M.
+21
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+Completeness.
+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. 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.e., pn,ω = 0 for each n ∈ N and ω ∈ Ω. Notice that the agents
+follow the recommendations since they are indifferent among all the actions. 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. This concludes the first part of the proof.
+Soundness.
+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.
+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. 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. Suppose by contradiction that there exists
+a contract with utility strictly larger than ρ. Then, there are strictly more than ρ|E| edges such that �
+σ∈Σ 1(ωnv,σ =
+1 ∧ ωnu,Πe(σ) = 1) = 1. Consider the assignment that assign to each variable v ∈ V ∪ U the label σ such that
+ˆanv = aσ. It is easy to see that this assignment satisfies strictly more than a ρ fraction of the edges, reaching a
+contradiction.
+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. 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. 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. 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
++ .
+Moreover, a reward function g : Rnq
++ → R is supermodular if its opposite function −g is submodular.
+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., 2017).
+Lemma 2. 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.
+Proof. The proof follows from Theorem 1 in (Østerdal, 2010). 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). 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. 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). This
+concludes the proof.
+Lemma 3. 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.
+Proof. 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. In particular, for
+every i ∈ N and j ∈ [ℓ], it holds πi(j) = (i, aj).
+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.
+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. 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′ .
+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. 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. Moreover, notice that, whenever a1,i = a2,i, it holds µi,ω
+0
+= 1 for
+every ω ∈ Ω. Then, we can write:
+Ra1 =
+�
+ω∈Ωn
+��
+i∈N
+Fi,a1,i,ωi
+�
+g(ω) =
+�
+ω∈Ω
+�
+i∈N
+� �
+ω′∈Ω
+Fi,a2,i,ω′µi,ω′
+ω−ω′
+�
+g(ω),
+RaS =
+�
+ω∈Ωn
+��
+i∈N
+Fi,aS,i,ωi
+�
+g(ω)
+=
+�
+ω∈Ωn
+
+
+
+�
+i∈N:
+aS,i=a1,i
+�
+ω′∈Ω
+Fi,a2,i,ω′µω′
+ω−ω′
+
+
+
+
+
+
+
+�
+i∈N:
+aS,i̸=a1,i
+Fi,a2,i,ωi
+
+
+
+ g(ω),
+and
+RaS′ =
+�
+ω∈Ωn
+��
+i∈N
+Fi,aS′,i,ωi
+�
+g(ω)
+=
+�
+ω∈Ωn
+
+
+
+
+�
+i∈N:
+aS′,i=a1,i
+�
+ω′∈Ω
+Fi,a2,i,ω′µω′
+ω−ω′
+
+
+
+
+
+
+
+
+�
+i∈N:
+aS′,i̸=a1,i
+Fi,a2,i,ωi
+
+
+
+ g(ω).
+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;
+• 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.
+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
+. 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). Let N1 := {i ∈ N : aS,i = a1,i}, and N2 := {i ∈ N : aS′,i = a1,i}. In order to conclude the proof,
+23
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+we show that the following holds:
+Ra1 + Ra2 =
+�
+ω∈Ωn
+��
+i∈N
+Fi,a1,i,ωi
+�
+g(ω) +
+�
+ω∈Ωn
+��
+i∈N
+Fi,a2,i,ωi
+�
+g(ω)
+(5a)
+=
+�
+ω∈Ωn
+��
+i∈N
+�
+ω′∈Ω
+Fi,a2,i,ω′µi,ω′
+ωi−ω′
+�
+g(ω) +
+�
+ω∈Ωn
+��
+i∈N
+Fi,a2,i,ωi
+�
+g(ω)
+(5b)
+=
+�
+ω∈Ωn
+�
+ω′∈Ωn
+��
+i∈N
+Fi,a2,i,ω′
+i
+� ��
+i∈N
+µi,ω′
+i
+ωi−ω′
+i
+�
+g(ω) +
+�
+ω∈Ωn
+��
+i∈N
+Fi,a2,i,ωi
+�
+g(ω)
+(5c)
+=
+�
+ω′∈Ωn
+��
+i∈N
+Fi,a2,i,ω′
+i
+� � �
+ω∈Ωn
+��
+i∈N
+µi,ω′
+i
+ωi−ω′
+i
+�
+g(ω) + g(ω′)
+�
+(5d)
+=
+�
+ω′∈Ωn
+��
+i∈N
+Fi,a2,i,ω′
+i
+� � �
+ω∈Ωn
+��
+i∈N
+µi,ω′
+i
+ωi−ω′
+i
+�
+g(ω) +
+�
+ω∈Ωn
+��
+i∈N
+µi,ω′
+i
+ωi−ω′
+i
+�
+g(ω′)
+�
+(5e)
+≥
+�
+ω′∈Ωn
+��
+i∈N
+Fi,a2,i,ω′
+i
+� �
+ω∈Ωn
+��
+i∈N
+µi,ω′
+i
+ωi−ω′
+i
+� �
+g(ωω,ω′
+1
+) + g(ωω,ω′
+2
+)
+�
+(5f)
+=
+�
+ω′∈Ωn
+��
+i∈N
+Fi,a2,i,ω′
+i
+�
+
+
+�
+ω∈Ωn:
+ωi=ω′
+i∀i∈N2
+�
+i∈N:
+aS,i=a1,i
+µi,ω′
+i
+ωi−ω′
+i g(ωω,ω′
+1
+)
++
+�
+ω∈Ωn:
+ωi=ω′
+i∀i∈N1
+�
+i∈N:
+aS′,i=a1,i
+µi,ω′
+i
+ωi−ω′
+i g(ωω,ω′
+2
+)
+
+
+(5g)
+=
+�
+ω∈Ωn
+
+
+
+
+�
+i∈N:
+aS,i̸=a1,i
+Fi,a2,i,ωi
+
+
+
+
+�
+ω′∈Ωn:
+ωi=ω′
+i∀i∈N2
+�
+i∈N:
+aS,i=a1,i
+Fi,a2,i,ω′
+i µi,ω′
+i
+ωi−ω′
+i g(ωω,ω′
+1
+)
++
+�
+ω′∈Ωn
+
+
+
+
+�
+i∈N:
+aS′,i̸=a1,i
+Fi,a2,i,ω′
+i
+
+
+
+
+�
+ω∈Ωn:
+ωi=ω′
+i∀i∈N1
+�
+i∈N:
+aS′,i=a1,i
+µi,ω′
+i
+ωi−ω′
+i g(ωω,ω′
+2
+)
+(5h)
+=
+�
+ω∈Ωn
+
+
+
+
+�
+i∈N:
+aS,i̸=a1,i
+Fi,a2,i,ωi
+
+
+
+
+�
+ω′∈Ωn:
+ωi=ω′
+i∀i∈N2
+�
+i∈N:
+aS,i=a1,i
+Fi,a2,i,ω′
+i µi,ω′
+i
+ωi−ω′
+i g(ω)
++
+�
+ω∈Ωn
+
+
+
+
+�
+i∈N:
+aS′,i̸=a1,i
+Fi,a2,i,ωi
+
+
+
+
+�
+ω′∈Ωn:
+ωi=ω′
+i∀i∈N1
+�
+i∈N:
+aS′,i=a1,i
+µi,ω′
+i
+ωi−ω′
+i g(ω)
+(5i)
+=
+�
+ω∈Ωn
+
+
+
+
+�
+i∈N:
+aS,i̸=a1,i
+Fi,a2,i,ωi
+
+
+
+
+
+
+
+�
+i∈N:
+aS,i=a1,i
+�
+ω′∈Ω
+Fi,a2,i,ω′
+i µi,ω′
+i
+ωi−ω′
+
+
+ g(ω)
++
+�
+ω∈Ωn
+
+
+
+
+�
+i∈N:
+aS′,i̸=a1,i
+Fi,a2,i,ωi
+
+
+
+
+
+
+
+
+�
+i∈N:
+aS′,i=a1,i
+�
+ω′∈Ω
+Fi,a2,i,ω′
+i µi,ω′
+i
+ωi−ω′
+
+
+
+ g(ω)
+(5j)
+= RaS + RaS′ ,
+(5k)
+24
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+where
+Equation
+(5b)
+follows
+from
+the
+definition
+of
+Ra1,
+Equation
+(5e)
+comes
+from
+the
+fact
+that
+�
+ω∈Ωn
+�
+i∈N µi,ω′
+i
+ωi−ω′
+i = 1, Equation (5f) holds since g(ω) + g(ω′) ≤ g(ωω,ω′
+1
+) + g(ωω,ω′
+2
+) by supermodular-
+ity, Equation (5g) follows from ωω,ω′
+1
+= ωω,ω′′
+1
+whenever ω′
+i = ω′′
+i for all i ∈ N1, ωω,ω′
+2
+= ωω,ω′′
+2
+whenever
+ω′
+i = ω′′
+i for all i ∈ N2, and �
+ω∈Ω µi,ω′
+ωi−ω′ = 1, Equation (5i) comes from ωω,ω′
+1
+= ωω,ω′′
+1
+whenever ω′
+i = ω′′
+i for
+all i ∈ N1 and ωω,ω′
+2
+= ωω,ω′′
+2
+whenever ω′
+i = ω′′
+i for all i ∈ N2, while Equation (5k) follows from the definition of
+RaS+ RaS′.
+Theorem 4. 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.
+Proof. 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. 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. Hence,
+Corollary 1 immediately provides a polynomial-time algorithm for finding a utility-maximizing contract.
+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.
+Definition 13 (GAP-ISα). 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.e., a subset of vertices such that there is no edge connecting two of
+them) of size at least |V |1−α;
+– all the independent sets have size at most |V |α.
+• Output: Determine which of the above two cases hold.
+GAP-ISα is known to be NP-hard for any α > 0 (H˚astad, 1999; Zuckerman, 2007).
+Now, we are ready to prove of Theorem 5.
+Proof of Theorem 5. Given a constant α > 0, we reduce from the problem GAP-ISα.
+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. 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. Moreover, we
+will see that in the multi-agent principal-agent problem in our reduction it holds n = |V |. Since GAP-ISα is NP-hard
+for each constant α > 0 this is sufficient to prove the statement.
+Construction.
+Given an instance of GAP-ISα G = (V, E), we build an instance of the multi-agent principal-agent
+problem as follows. For each vertex v ∈ V , there exists an agent nv with actions a1 and a0. The outcome space is
+given by R+. Then, for each agent v ∈ V action a1 induces deterministically outcome ω1 = 1, i.e., Fnv,a1,ω1 = 1.
+Moreover, action a1 has cost 1 − δ, i.e., cnv,a1 = 1 − δ, where δ =
+1
+|V |2 . For each agent v ∈ V action a0 induces
+deterministically outcome ω0 = 0, i.e., Fnv,a0mω0 = 1. Moreover, action a0 has cost 0, i.e., cnv,a0 = 0. Given a node
+v ∈ V , let kv ≤ |V | be the degree of node v. Finally, the utility function g is defined as
+g(ω) =
+�
+(u,v)∈E
+max{ 1
+ku
+ωu, 1
+kv
+ωv}.
+Notice that the function is DR-submodular since it is the sum of DR-submodular functions. 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.
+25
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+Completeness.
+Suppose that there exists an independent set V ∗ ⊆ V of G with size at least |V |1−α. 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. Finally, we recommend to all the agent nv, v ∈ V ∗, to play a1, i.e., anv = a1, and to all the agents nv,
+v ∈ V \ V ∗ to play a0, i.e., anv = a0. It is easy to see that the action profile a is such that ai is IC under p for each
+agent i ∈ N. 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 ∗.
+Moreover, the total payment is given by �
+v∈V ∗(1 − δ) = |V ∗|(1 − δ). Thus, the total principal’s utility is given by
+|V ∗| − (1 − δ)|V ∗| = δ|V ∗|.
+Soundness.
+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. First, we show that if the contract incentivizes two
+agents nu and nv with (u, v) ∈ E, i.e., u and v are adjacent vertexes, to play action a1, then the principal’s utility is
+negative. Let ¯V be the set of nodes relative to agents incentivized to play a1, i.e., the set of i ∈ N such that ai = a1,
+and ¯E be the set of edges connecting two nodes in ¯V . 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 |.
+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. Hence,
+the principal’s utility is at most | ¯V | − 1/|V | − (1 − δ)| ¯V | = δ| ¯V | − 1/|V | < 0.
+Hence, in any contract with positive utility there are not two agents nv, nu relative to adjacent vertexes, i.e., such that
+(v, u) ∈ E playing action a1. Since all the independent sets has size at most |V |α, this implies that | ¯V | ≤ |V |α. Then,
+the reward of the contract is given by �
+v∈ ¯V
+�
+u:(u,v)∈E
+1
+kv 1 = | ¯V |. Moreover, the payment is at least (1 − δ)| ¯V |
+since for each agent nv, v ∈ ¯V it holds pnv,ω1 ≥ 1 − δ. However, the principal’s utility is at most | ¯V | − (1 − δ)| ¯V | =
+δ| ¯V | ≤ δ|V |α.
+E
+Proofs Omitted from Section 5
+Lemma 4. 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.
+26
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+Proof. 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).
+It is easy to check that fI is monotone-increasing, since the reward function g is increasing by assumption (see As-
+sumption 2). Thus, we are left to show that fI is also submodular.
+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. Moreover, given two tuples
+ω, ω′ ∈ Ωn, we let ω + ω′ be the tuple whose i-th outcome is ωi + ω′
+i.
+In order to prove that fI is submodular, we need to show that, for any two subsets S ⊂ S′ ⊆ GI and element
+(i, a) ∈ GI, it holds fI(S ∪ {(i, a)}) − fI(S) ≤ fI(S′ ∪ {(i, a)}) − fI(S′):
+fI(S′ ∪ {(i, a)}) − fI(S′) = RS∪{(i,a)} − RS′
+=
+�
+ω∈˜Ωn
+rω
+�
+j∈N
+Fj,S′∪{(i,a)},ωj −
+�
+˜ω∈Ωn
+rω
+�
+j∈N
+Fj,S′,ωj
+=
+�
+ω∈˜Ωn
+�
+ω′∈˜Ωn
+�
+ω∈Ω
+rω+ω′+ωi,ω Fi,a,ω
+
+ �
+j∈N
+Fj,S,ωj
+
+
+
+ �
+j∈N
+Fj,S′\S,ωj
+
+
+−
+�
+ω∈˜Ωn
+�
+ω′∈˜Ωn
+rω+ω′
+
+ �
+j∈N
+Fj,S,ωj
+
+
+
+ �
+j∈N
+Fj,S′\S,ωj
+
+
+=
+�
+ω∈˜Ωn
+�
+ω′∈˜Ωn
+�
+ω∈Ω
+Fi,a,ω
+
+ �
+j∈N
+Fj,S,ωj
+
+
+
+ �
+j∈N
+Fj,S′\S,ωj
+
+ (rω+ω′+ωi,ω − rω+ω′)
+≤
+�
+ω∈˜Ωn
+�
+ω′∈˜Ωn
+�
+ω∈Ω
+Fi,a,ω
+
+ �
+j∈N
+Fj,S,ωj
+
+
+
+ �
+j∈N
+Fj,S′\S,ωj
+
+ (rω+ωi,ω − rω)
+=
+�
+ω∈˜Ωn
+�
+ω∈Ω
+Fi,a,ω
+
+ �
+j∈N
+Fj,S,ωj
+
+ �
+rω+1i(¯ω) − rω
+�
+=
+�
+ω∈˜Ωn
+�
+ω∈Ω
+Fi,a,ω
+
+ �
+j∈N
+Fj,S,ωj
+
+ rω+ωi,ω −
+�
+ω∈˜Ωn
+
+ �
+j∈N
+Fj,S,ωj
+
+ rω
+=
+�
+ω∈˜Ωn
+rω
+�
+j∈N
+Fj,S∪{(i,a),ωj} −
+�
+ω∈˜Ωn
+rω
+�
+j∈N
+Fj,S,ωj
+= fI(S ∪ {(i, a)}) − fI(S),
+where the inequality hold by DR-submodularity. This concludes the proof.
+Theorem 6. 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∗).
+Proof. 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., 2017).
+In particular, Theorem 3.1 in (Sviridenko et al., 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})|}.
+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. 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.
+To do that, let ˜GI ⊆ GI be the set of elements (i, a) such that �Pi,a > 1. 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]). Hence, we
+can optimize over the matroid that only includes the elements in GI \ ˜GI, so that we get ˆv ≤ 1. This concludes the
+proof.
+F
+Proofs Omitted from Section 6
+Lemma 5. 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.e.,
+the agent i’s payment scheme under contract p).
+Proof. 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. Let ¯p be such that ¯pj = pj for each j ̸= i and
+¯pi = ˆγˆθ
+pipi + ˆγˆθ
+p′
+ip′
+i. Moreover, let ¯p′ be such that ¯p′
+j = p′
+j for each j ̸= i and ¯p′
+i = ˆγˆθ
+pipi + ˆγˆθ
+p′
+ip′
+i. 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}.
+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.
+Moreover, γθ for each θ ∈ Θn provides the same utility of ˆγθ for each θ ∈ Θn. To conclude the proof we need to
+proof that γθ for each θ ∈ Θn is DSIC. Following an analysis similar to the one in Castiglioni et al. (2022a) we can
+show that replacing the marginal contracts pi and p′
+i with the weighted combination ¯pi the DSIC constraint continue
+to hold. Applying this operation until such two contracts does not exist is sufficient to prove the statement.
+Lemma 6. 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.20
+Proof. 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|. 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.g., (Bertsimas and Tsitsiklis, 1997)). 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.
+Lemma 7. For every instance of Bayesian principal-multi-agent problem, it holds LP ≥ SUP.
+20In the rest of the section, we always assume that the size of a problem instance is expressed in terms of number of bits.
+28
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+Proof. 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.
+Let (tθ,a, ξi,θ,a, pi,θ,a,ω) be a feasible solution to Problem (1). 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. 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). 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.
+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.
+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). 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.
+Lemma 8. 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.
+Proof. Let (tθ,a, ξi,θ,a, yi,θ,a,ω, γi,θ,θ,,a) be a feasible solution to LP (2). 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.e., the tuples of indexes identifying the pairs of variables
+that do not meet regularity conditions).
+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. 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)). Then, let us define yw
+i,θ,a,ω = pw
+i,ω for all ω ∈ Ω, while
+ξw
+i,θ,a = 1. 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. Finally, we let tw
+θ,a = 1. 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|).
+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
+�
+.
+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. Moreover, it can be computed in
+time polynomial in |I| and 1
+ǫ, concluding the proof.
+Theorem 7. 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.
+Proof. 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. Thus, if the given optimal solution to LP (2) is regular, then the result
+immediately follows by Lemma 7, since LP = SUP. 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.
+29
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+Lemma 9. For every instance of Bayesian principal-multi-agent problem, LP (2) and LP (3) have the same optimal
+value. 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.
+Proof. 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.
+Let (tθ,a, ξi,θ,a, yi,θ,a,ω, γi,θ,θ,,a) be a feasible solution to LP (3).
+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. 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. Moreover, let δθ := 1 − �
+a∈An,θ tθ,a. 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.
+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.e., for every i ∈ N and a ∈ Ai,θi, it holds �
+a∈An,θ:ai=a ¯tθ,a = δi,θ,a/δθ.
+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. 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. Then, let us define
+ˆtθ,a = δθ¯tθ,a for every a ∈ An,θ. 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.
+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. 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).
+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. 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. Thus, such a solution is also feasible for LP (2).
+Moreover, it is easy to see that it can be computed in polynomial time.
+Theorem 8. 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. Moreover, such an algorithm runs in time polynomial in the instance size and 1
+ρ.
+Proof. We start by providing the general procedure underlining the approximation algorithm (see Algorithm 1). 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. The constant β ≥ 0 will be specified later in the proof.
+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.
+F
+
+
+
+�
+i∈N
+�
+θ∈Θ
+�
+θ−i∈ ˜Θn
+−i
+xi,θ,θ−i ≤ η
+Constraints (4b)—(4i).
+The algorithm is initialized with l = 0, h = 1. 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). If
+F is found to be infeasible by the ellipsoid method, the algorithm sets l ← η. Otherwise, if
+F is found
+to be “approximately” feasible, the algorithm sets h ← η. 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.21
+In the following, we first describe in details the ad hoc implementation of the ellipsoid method employed by Al-
+gorithm 1. 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. Finally, we put all the bounds together in order to prove the statement of the theorem.
+Algorithm 1 Approximation algorithm introduced in the proof of Theorem 8
+Input: Bayesian principal-multi-agent problem instance I := (N, Θ, Ω, A); Multiplicative approximation factor
+α ∈ (0, 1]; Additive approximation error ρ > 0; Approximate separation oracle Oα(·, ·, ·, ·) for Constraints (4e)
+1: Initialization: l ← 0; h ← 1; H ← ∅; H⋆ ← ∅; β ← ρ
+4; 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; H⋆ ← H
+8:
+else
+9:
+h ← η
+10: return η⋆ ← h; Optimal solution to LP (3) in which only the variables tθ,a corresponding to the dual constraints
+in H⋆ are specified
+Implementation of the ellipsoid method
+Given 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) and η ∈ [0, 1], our implementation of the ellipsoid method employs an ad hoc separation oracle to determine
+whether the point (xi,θ,θ−i, yi,θ,a, zi,θ,θ,a,a′, di,θ,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. In the latter case, the separation oracle returns the
+violated constraint.
+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. If a violated
+constraint is found, the oracle returns it.
+If all the Constraints (4b)—(4d) and the Constraints (4f)—(4i) are not violated, the oracle has to check the
+exponentially-many Constraints (4e). 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. 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. 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. Instead, if for some θ ∈ ˜Θn
+the call Oα(I, w, θ, ǫ) returns an action profile a ∈ An,θ such that λθRθ,a − �
+i∈N wi,ai > 0, then it must be
+21Notice that, in the case in which the ad hoc implementation of the ellipsoid method concludes that
+F is “approximately”
+feasible at every iteration, or, similarly, when it always returns infeasible, we can perform a similar analysis by observing that there
+always exists a feasible solution with objective 1, while all the feasible solutions have objective at least 0.
+31
+
+ARXIV PREPRINT - FEBRUARY 1, 2023
+wi,ai ≤ 1 for all i ∈ N (as rewards belong to [0, 1]). 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. 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.
+Bounding the value of an optimal solution
+Next, prove that αROPT − POPT − |˜Θn|ǫ ≤ η⋆ for any optimal DSIC
+menu of randomized contracts.
+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. 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.
+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⋆.
+min
+�
+i∈N
+�
+θ∈Θ
+�
+θ−i∈ ˜Θn
+−i
+xi,θ,θ−i
+s.t.
+(6a)
+α λθ Rθ,a −
+�
+i∈N
+yi,θ,ai ≤ ǫ
+∀θ ∈ ˜Θn, ∀a ∈ An,θ
+(6b)
+Constraints (4b)—(4d) and (4f)—(4i).
+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. 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). 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. This shows that (xi,θ,θ−i, yi,θ,a, zi,θ,θ,a,a′, di,θ,θ−i) is feasible for LP (6).
+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.t.
+(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).
+By strong duality, we have that the optimal value of LP (7) is at most η⋆. 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. This proves that
+αROPT − POPT − |˜Θn|ǫ ≤ η⋆.
+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.
+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. 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.
+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.
+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. 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.t.
+(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).
+By strong duality, LP (9) has optimal value at least η⋆ − β. 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.
+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. Let APX be the value of an optimal solution to LP (9). Moreover, set β := ρ
+4
+and ǫ :=
+ρ
+4| ˜Θn|. Then,
+APX ≥ η⋆ − β
+≥ αROPT − POPT − |˜Θn|ǫ − β
+≥ αROPT − POPT − ρ
+2.
+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)). 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.
+Corollary 3. 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.
+Proof. We show that the problem admits a polynomial-time approximate separation oracle O1(·, ·, ·, ·). Then, the
+result directly follows from Theorem 8.
+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). 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.
+It is easy to see that the arguments proving Theorem 4 continue to hold.
+Corollary 4. In Bayesian principal-multi-agent problem instances with succinct rewards specified by a DR-
+submodular function, the problem of computing an optimal menu of randomized contracts admits a polynomial-time
+approximation algorithm which, for any ǫ > 0 given as input, 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,
+where RΓ ∈ [0, 1], respectively PΓ ∈ R+, denotes the expected reward, respectively the expected payment, in contract
+p.
+Proof. We show that the problem admits a polynomial-time approximate separation oracle O1−1/e(·, ·, ·, ·). Then, the
+result readily follows from Theorem 8.
+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). 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. It is easy
+to see that Theorem 6 continues to hold.
+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.
+33
+