diff --git "a/3tFST4oBgHgl3EQfZDim/content/tmp_files/load_file.txt" "b/3tFST4oBgHgl3EQfZDim/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/3tFST4oBgHgl3EQfZDim/content/tmp_files/load_file.txt" @@ -0,0 +1,1340 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf,len=1339 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='13790v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='GT] 31 Jan 2023 SELLING INFORMATION WHILE BEING AN INTERESTED PARTY ARXIV PREPRINT Matteo Castiglioni Politecnico di Milano matteo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='castiglioni@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='it Francesco Bacchiocchi Politecnico di Milano francesco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='bacchiocchi@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='it Alberto Marchesi Politecnico di Milano alberto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='marchesi@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='it Giulia Romano Politecnico di Milano giulia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='romano@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='it Nicola Gatti Politecnico di Milano nicola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='gatti@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='it February 1, 2023 ABSTRACT We study the algorithmic problem faced by an information holder (seller) who wants to optimally sell such information to a budged-constrained decision maker (buyer) that has to undertake some action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Differently from previous works addressing this problem, we consider the case in which the seller is an interested party, as the action chosen by the buyer does not only influence their utility, but also seller’s one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This happens in many real-world settings, where the way in which businesses use acquired information may positively or negatively affect the seller, due to the presence of externalities on the information market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The utilities of both the seller and the buyer depend on a random state of nature, which is revealed to the seller, but it is unknown to the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, the seller’s goal is to (partially) sell their information about the state of nature to the buyer, so as to concurrently maximize revenue and induce the buyer to take a desirable action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We study settings in which buyer’s budget and utilities are determined by a random buyer’s type that is unknown to the seller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In such settings, an optimal protocol for the seller must propose to the buyer a menu of information-revelation policies to choose from, with the latter acquiring one of them by paying its corresponding price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, since in our model the seller is an interested party, an optimal protocol must also prescribe the seller to pay back the buyer contingently on their action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First, we show that the problem of computing a seller-optimal protocol can be solved in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This result relies on a quadratic formulation of the problem, which we solve by means of a linear programming relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Next, we switch the attention to the case in which a seller’s protocol employs a single information-revelation policy, rather than proposing a menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In such a setting, we show that computing a seller-optimal protocol is APX-hard, even when either the number of actions or that of states of nature is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We complement such a negative result by providing a quasi- polynomial-time approximation algorithm that, given any ρ > 0 and ǫ > 0 as input, provides a multiplicative approximation ρ of the optimal seller’s expected utility, by only suffering a negligible 2−Ω(1/ρ) + ǫ additive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Such an algorithm runs in polynomial time whenever either the number of buyer’s actions or that of states of nature is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In order to derive our results, we draw a connection between our information-selling problem and principal-agent problems with observable actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, we complete the picture of the computational complexity of finding seller-optimal protocols without menus by providing additional results for the specific setting in which the buyer has limited liability, and by designing a polynomial-time algorithm for the case in which buyer’s types are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ARXIV PREPRINT - FEBRUARY 1, 2023 1 Introduction Nowadays, there is a terrific amount of information being collected on the Web and other online platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Such infor- mation ranges from consumer preferences, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', in e-commerce and streaming websites, to credit reports and location histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a result, recent years have witnessed the born and exponential blowout of markets where specialized companies sell information that is valuable to other businesses, such as advertisers, retailers, and loan providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Very recently, information markets have also received the attention of the algorithmic game theory research commu- nity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' However, while works addressing classical settings such as auctions (Daskalakis and Syrgkanis, 2022), signal- ing (Dughmi and Xu, 2019), and contract design (Dütting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2019) are now proliferating, only few papers studied the problem of information selling, with (Babaioff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2012) and (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2020) constituting two notable exam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We study the algorithmic problem faced by an information holder (seller) who wants to optimally sell such information to a budged-constrained decision maker (buyer) that has to undertake some action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Differently from previous works addressing such a problem (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2020)), we consider the case in which the seller is an interested party, as the action chosen by the buyer does not only influence their utility, but also seller’s one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This happens in many real-world settings, where the way in which businesses use acquired information may positively or negatively affect the seller, due to the presence of externalities on the information market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The utilities of both the seller and the buyer depend on a state of nature that is drawn according to a commonly-known probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The realized state of nature is revealed to the seller, while it remains unknown to the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, the seller’s goal is to (partially) sell their information about the state of nature to the buyer, so as to concurrently maximize revenue and induce the buyer to take a desirable action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We study settings in which buyer’s budget and utilities are determined by a random buyer’s type that is unknown to the seller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In such settings, in order to optimally sell information, the seller has to commit upfront to a protocol working as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First, the seller proposes to the buyer a menu of information-revelation policies to choose from, and the latter acquires an expected-utility-maximizing one according to their (private) type, by paying its corresponding price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By building on the Bayesian persuasion framework introduced by Kamenica and Gentzkow (2011), an information- revelation policy is implemented as a signaling scheme, which is a randomized mapping from states of nature to signals issued to the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, the realized state of nature is disclosed to the seller, who reveals information about it to the buyer according to the acquired signaling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, the buyer selects a best-response action according to the just acquired information, and the seller pays back the buyer with a payment which depends on both the chosen action and the signal that has been previously sent by the seller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Our protocol extends the one of Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2020) by adding a final payment from the seller to the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As we show later, this is crucial in order to design seller-optimal protocols in our setting where the seller is an interested party, since the latter is not only concerned with revenue, but also with the buyer’s action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the addition of payments from the buyer to the seller is also reasonable in many real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For instance, think of a case in which the information holder asks the buyer to deposit additional money, and this is given back to them only if the performed action respects some given rules on which the two parties agreed upfront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='1 Original Contributions After introducing all the needed concepts in Section 2, we start providing our results in Section 3, where we analyze the case of general protocols in which the seller proposes a menu of signaling schemes to the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that a seller-optimal protocol can be computed in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In order to do that, we first formulate the problem of finding a seller-optimal protocol as a quadratic problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, we show that one can focus on direct and persuasive signaling schemes, which are those that send signals corresponding to action recommendations for the buyer and properly incentivize the latter to follow such recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This in turn allows us to restrict the attention to protocols that ask the buyer to pay their entire budget upfront and, then, pay back the buyer only if they take the recommended action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' These results allow us to formulate a suitable linear relaxation of the quadratic problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A similar technique has been employed in generalized principal-agent problems (Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2022), where it is possible to show that an optimal solution to the linear relaxation can be efficiently cast to an approximately-optimal solution to the quadratic problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022b) show that, even in the special case of hidden-action principal- agent problems, obtaining an optimal solution to the quadratic problem is not possible in general, since the principal’s optimization problem may not admit a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Surprisingly, in our information-selling setting, we prove that an optimal solution to our linear relaxation, which can be computed in polynomial time, can be used to recover a seller- optimal protocol in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a byproduct, this also shows that, in our setting, the seller’s problem always admits a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2 ARXIV PREPRINT - FEBRUARY 1, 2023 In the second part of the paper, we switch the attention to the case of protocols without menus, in which the seller does not propose a menu of signaling schemes to the buyer, but they rather commit to a single signaling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This is the case in many real-world applications, where it is unreasonable that a buyer is asked to choose an information-revelation policy among a range of options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Computing a seller-optimal protocol without menus begets considerable additional computational challenges, since, intuitively, the seller has no way of extracting information about the buyer’s private type, as instead it is the case when proposing a menu to choose from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In Section 4, we draw a connection between the problem of computing a seller-optimal protocol without menus and principal-agent problems with observable actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' These are problems in which a principal commits to an action- dependent payment scheme in order to incentivize an agent to take some costly, observable action, in order to maximize their expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We prove that observable-action principal-agent problems are a special case of our information- selling problem, and that, in such problems, computing an expected-utility-maximizing payment-scheme for the prin- cipal is APX-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In particular, these results show that our information-selling problem is APX-hard even when the number of states of nature is fixed and the buyer has limited liability, and, thus, the seller cannot charge a price for a signaling scheme upfront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We also provide some preliminary technical results on observable-action principal-agent problems, which are useful in order to prove some of our main claims in the paper, while also being of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In Section 5, we show how to circumvent the APX-hardness for settings in which the seller employs protocols without menus and the buyer has limited liability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' These special settings are of interested on their own, as a similar model has been recently addressed by Dughmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We focus on special cases where one of the parameters characterizing a problem instance is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In particular, we study what happens if we fix the number of buyer’s actions, showing that the problem admits a PTAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, we prove that, when instead the number of states of nature is fixed, there exists a polynomial-time bi-criteria approximation algorithm that, given any ρ > 0 and ǫ > 0 as input, provides a multiplicative approximation ρ of the optimal seller’s expected utility, by only suffering a 2−Ω(1/ρ) + ǫ additive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that such a loss is exponentially small in 1 ρ, and, thus, it is negligible even for reasonably large values of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As shown by Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022a), such an approximation result is tight for hidden-action principal-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It remains an open problem to establish whether such an approximation guarantee is also tight for principal-agent problems with observable actions, which are a special case of our information-selling problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Table 1: Summary of the results provided in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Each cell specifies, on the first line, the computational com- plexity of finding a seller-optimal protocol, while, additionally, on the second line, it specifies the approximation guarantees that we can obtain in polynomial time, where OPT denotes the seller’s expected utility in an optimal proto- col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The approximation guarantees that are shaded in gray can only be obtained by means of a quasi-polynomial-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' general fixed # actions fixed # states fixed # types Protocols with menus P P P P Protocols w/o menus Buyer w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' limited liability APX-hard — APX-hard P ρOPT−2−Ω(1/ρ)−ǫ PTAS ρOPT−2−Ω(1/ρ)−ǫ Protocols w/o menus Buyer w/o limited liability APX-hard APX-hard APX-hard P ρOPT−2−Ω(1/ρ)−ǫ ρOPT−2−Ω(1/ρ)−ǫ ρOPT−2−Ω(1/ρ)−ǫ In conclusion, in Section 6 we study the problem of computing seller-optimal protocols without menus in general settings in which the buyer does not have limited liability, and, thus, the seller can charge a price for a signaling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We first prove a stronger negative result, by showing that, in such a setting, the problem of computing a seller- optimal protocol is APX-hard even if the number of buyer’s actions is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, we show how to circumvent such a negative result by providing a quasi-polynomial-time bi-criteria approximation algorithm that, given any ρ > 0 and ǫ > 0 as input, provides a multiplicative approximation ρ of the optimal seller’s expected utility, plus a2−Ω(1/ρ) + ǫ additive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We prove that, when either the number of buyer’s action or that of states of nature is fixed, such an algorithm runs in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, we show that, when the number of buyer’s types is fixed, the problem admits a polynomial-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This also implies that the seller’s optimization problem for protocols without menus always admits a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We summarize the results provided in this paper in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' All the proofs are in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 3 ARXIV PREPRINT - FEBRUARY 1, 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2 Related Works The study of algorithmic ways of selling information to an imperfectly-informed buyer has received some attention in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Babaioff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2012) initiated the study by considering a buyer with an unlimited budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' They provide an exponentially-sized linear program (LP) for computing an optimal mechanism for selling information, and they efficiently solve it through the ellipsoid method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The main drawback of the approach presented by Babaioff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2012) is that an optimal mechanism may require a significant money transfer from the buyer to the seller and viceversa, in order to only achieve a small, overall net transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2020) complement the results in (Babaioff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2012) by studying the problem of selling information when both the buyer and the seller are budget-constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, they also consider a setting in which the buyer’s budget is private, and the seller needs to elicit it in the mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2020) show that the addition of budget constraints considerably simplifies the problem of computing an optimal mechanism, since it can be formulated as a polynomially-sized LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The problem of selling information has also been addressed by Bergemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2018), who study the case of bi- nary actions and states of nature, characterizing a revenue-maximizing mechanism in such a setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Furthermore, Bergemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022) extend the analysis to the case in which there are more than two actions and binary states of nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In contrast, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2021) study a revenue-maximizing mechanism for selling information when the stochasticity of the state of nature only affects a subset of the actions of the decision maker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Our problem is also related to the Bayesian persuasion framework originally introduced by Kamenica and Gentzkow (2011), where an informed sender wants to influence the behavior of a self-interested receiver via the strategic provision of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Dughmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2019) generalize the classical framework by considering the case in which there are monetary transfers between the sender and the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Our information-selling setting in which the buyer has limited liability generalizes the model of Dughmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2019) by also introducing buyer’s types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, let us remark that our information-selling problem shares critical features with Bayesian principal-agent prob- lems (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', (Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Alon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2021, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Guruganesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2022c) for some references).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, as we show in Section 4, the problem of computing a seller-optimal protocol gener- alizes particular principal-agent problems in which the agent’s action is observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Such a connection between the two settings is also demonstrated in terms of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In particular, notice that Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022a) design bi- criteria approximation algorithms whose guarantees are similar to those provided in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022) show how to find optimal protocols in generalized principal-agent problems by using a linear relaxation of the principal’s optimization problem, which is quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2 Preliminaries We study the problem faced by an information holder (seller) selling information to a budget-constrained decision maker (buyer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The information available to the seller is collectively termed state of nature and encoded as an element of a finite set Θ := {θi}d i=1 of d possible states, while the set of the m actions available to the buyer is A := {ai}m i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The buyer is also characterized by a private type, which is unknown to the seller and belongs to a finite set K := {ki}n i=1 of n possible types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Each buyer’s type k ∈ K is characterized by a utility function uk θ : A → [0, 1] associated to each state θ ∈ Θ and a budget bk ∈ R+ representing how much they can afford to pay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In our model, the seller’s utility is not only determined by how much the buyer pays for acquiring information, but it also depends on the buyer’s action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Specifically, for every state θ ∈ Θ, the sender gets an additional utility contribution determined by a function us θ : A → [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We assume that both the seller and the buyer know the probability distribution µ ∈ ∆Θ according to which the state of nature is drawn, as well as the probability distribution λ ∈ ∆K determining the buyer’s type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='1 We let µθ be the probability assigned to state θ ∈ Θ, while λk is the probability of type k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As in (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2020), we assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' that information revelation happens only once during the seller-buyer interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, as it is the case in Bayesian persuasion Kamenica and Gentzkow (2011), the seller reveals infor- mation to the buyer by committing to a signaling scheme φ, which is a randomized mapping from states of nature to signals being issued to the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, φ : Θ → ∆S, where S is a finite set of signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We denote by φθ ∈ ∆S the probability distribution employed when the state of nature is θ ∈ Θ, with φθ(s) being the probability of sending s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='1 Protocols with Menus An information-selling protocol for the seller is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The seller first proposes a menu of signaling schemes to the buyer, with each signaling scheme being assigned with a price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, the buyer chooses a signaling 1In this work, given a finite set X, we let ∆X be the set of all the probability distributions defined over the elements of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 4 ARXIV PREPRINT - FEBRUARY 1, 2023 scheme and pays its price upfront, before information is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2 The seller also commits to action-dependent payments, which are made by the seller in favor of the buyer after information is revealed and the latter has taken an action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This is in contrast with what happens in the protocol introduced by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2020), where there are no action-dependent money transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, such payments are needed in order to incentivize the agent to play an action that is profitable for the seller, and, thus, they are not needed in the setting of Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2020) where the seller’s utility function is only determined by how much the buyer pays for acquiring information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, we define a seller’s protocol as follows: Definition 1 (Seller’s protocol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A protocol for the seller is a tuple {(φk, pk, πk)}k∈K, where: {φk}k∈K is a menu of signaling schemes φk : Θ → ∆S, one for each receiver’s type k ∈ K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' {pk}k∈K is a menu of prices, with pk ∈ R+ representing how much the seller charges the buyer for selecting the signaling scheme φk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='3 {πk}k∈K is a menu of payment functions, which are defined as πk : S × A → R+ with πk(s, a) encoding how much the seller pays the buyer whenever the latter plays action a ∈ A after selecting the signaling scheme φk and receiving signal s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='4 The seller and the buyer interact as follows: (i) the seller commits to a protocol {(φk, pk, πk)}k∈K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (ii) the buyer selects a signaling scheme φk and pays pk to the seller (with k ∈ K possibly different from their true type);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (iii) the seller observes the realized state of nature θ ∼ µ, draws a signal s ∼ φk θ according to the selected signaling scheme, and communicates s to the buyer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (iv) given the signal s, the buyer infers a posterior distribution ξs ∈ ∆Θ over states of nature, where the probability ξs θ of state θ ∈ Θ is computed with the Bayes rule, as follows: ξs θ := µθ φk θ(s) � θ′∈Θ µθ′φk θ′(s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (v) given the posterior ξs, the buyer selects an action a ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' and (vi) the seller pays πk(s, a) to the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As in the model by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2020), we assume that the seller is committed to following the protocol, while the buyer is not, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', the buyer is free of leaving the interaction at any point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In step (v), after observing a signal s ∈ S and computing the posterior ξs, the buyer plays a best response by choosing an action a ∈ A maximizing their expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: Definition 2 (ǫ-Best-response).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let ǫ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a signal s ∈ S, the induced posterior ξs ∈ ∆Θ, and a payment function π : S × A → R+, the ǫ-best-response set of a buyer of type k ∈ K is: Bk,ǫ ξs,π := � a ∈ A : � θ∈Θ ξs θ uk θ(a) + π(s, a) ≥ max a′∈A � θ∈Θ ξs θ uk θ(a′) + π(s, a′) − ǫ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We let bk,ǫ ξs,π ∈ Bk,ǫ ξs,π be an ǫ-best response played by the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The best-response set Bk ξs,π of a buyer of type k ∈ K is defined for ǫ = 0, while bk ξs,π ∈ Bk ξs,π is a best response played by the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='5 In the following, we will oftentimes work in the space of the distributions over posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In that case, given a posterior ξ ∈ ∆Θ, we abuse notation and write Bk ξ,π, Bk,ǫ ξ,π, bk,ǫ ξ,π, and bk ξ,π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The seller’s goal is to implement an optimal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', utility-maximizing) protocol {(φk, pk, πk)}k∈K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We focus on seller’s protocols that are incentive compatible (IC) and individually rational (IR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='6 Specifically, a seller’s protocol is IC if for every pair of buyer’s types k, k′ ∈ K: � s∈S � θ∈Θ µθφk θ(s) � uk θ(bk ξs,πk) + πk(s, bk ξs,πk) � − pk ≥ � s∈S max a∈A � θ∈Θ µθφk′ θ (s) � uk θ(a) + πk′(s, a) � − pk′, 2Notice that proposing a menu of signaling schemes is equivalent to asking the buyer to report their type and then choosing a signaling scheme based on that, as it is the case in (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 3Assuming pk ≥ 0 is w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', since, intuitively, the seller is never better off paying the buyer before they played any action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 4The assumption that πk(s, a) ≥ 0 is w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', since the buyer does not commit to following the protocol, and, thus, πk(s, a) < 0 would result in the buyer leaving the protocol without paying after taking an action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 5When the buyer is indifferent among multiple best responses (respectively, ǫ-best responses), we always assume that they break ties in favor of the seller, choosing an action in Bk ξs,π (respectively, Bk,ǫ ξs,π) maximizing the seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 6By a revelation-principle-style argument (see (Shoham and Leyton-Brown, 2008) for some examples), focusing on IC and IR protocols is w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' when looking for an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 5 ARXIV PREPRINT - FEBRUARY 1, 2023 while it is IR if for every buyers’ type k ∈ K: � s∈S � θ∈Θ µθφk θ(s) � uk θ(bk ξs,πk) + πk(s, bk ξs,πk) � − pk ≥ max a∈A � θ∈Θ µθuk θ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, an IC protocol incentivizes the buyer to select the signaling scheme φk corresponding ot their true type k ∈ K, while an IR protocol ensures that the buyer gets more utility by acquiring information rather than leaving the protocol before step (ii) and playing an action without information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, the seller’s expected utility is computed as follows: � k∈K λk �� s∈S � θ∈Θ µθφk θ(s) � us θ(bk ξs,πk) − πk(s, bk ξs,πk) � + pk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A crucial component of our results is that we can restrict the attention to protocols that are direct and persuasive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We say that protocol is direct if it uses signaling schemes whose signals correspond to action recommendations for the buyer, namely S = A, while a direct protocol is said to be persuasive whenever playing the recommended action is always a best response for the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2 Protocols without Menus In the second part of the paper, we study the case of seller’s protocols without menus, in which the seller does not propose a menu of signaling schemes to the buyer, but they rather commit to a single signaling scheme and a single payment function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='7 This allows us to simplify the definition of a protocol (see Definition 1), by denoting a seller’s protocol without menus as a tuple (φ, p, π), where φ : Θ → ∆S is a signaling scheme, p ∈ R+ is a price for such a signaling scheme, representing how much the seller charges the buyer to reveal information to them, and π : S × A → R+ is a payment function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The seller-buyer interaction unfolds as in the general case with menus, but, in this case, step (ii) only involves the payment of price p ∈ R+ on buyer’s part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Some of our results on protocols without menus address the special case in which the buyer has limited liability, which means that the buyer has no budget, and, thus, the seller cannot charge a price for a signaling scheme upfront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, this amounts to asking that bk = 0 for all k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that, while such a special case may seem of scarce appeal for the problem of selling information, it is indeed interesting on its own, as it is similar to the model studied by Dughmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, our model can be seen as a generalization of the one in (Dughmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2019), which adds buyer’s private types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, in the general case in which the buyer has no limited liability, our model additionally builds on top of that of Dughmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2019) by adding the possibility for the seller to ask the buyer a payments before information is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For protocols without menus, IC constraints are not needed anymore, while IR constraints are still required in order to ensure that the buyer is incentivized to acquire information from the principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a protocol without menus (φ, p, π), only some of the buyer’s types are actually incentivized to participate in the protocol, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', all the types whose corresponding IR constraint is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, a protocol determines a subset Rφ,p,π ⊆ K of buyer’s types such that, for every k ∈ Rφ,p,π, it holds that: (i) a buyer of type k has enough budget to buy information, namely bk ≥ p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' and (ii) the IR constraint is satisfied for a buyer of type k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='8 In particular, point (ii) can be formally stated by saying that the following condition is satisfied for every k ∈ Rφ,p,π: � s∈S � θ∈Θ µθφθ(s) � uk θ(bk ξs,π) + π(s, bk ξs,π) � − p ≥ max a∈A � θ∈Θ µθuk θ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, given a protocol without menus (φ, π, p), the seller’s expected utility is given by: � k∈Rφ,p,π λk �� s∈S � θ∈Θ µθφθ(s) � us θ(bk ξs,π) − π(s, bk ξs,π) � + p � + � k̸∈Rφ,p,π λk � θ∈Θ µθuk θ(bk µ), where bk ξ ∈ arg maxa∈A � θ∈Θ ξθuk θ(a) is a best response for a buyer’s type k ∈ K that only considers the posterior ξ ∈ ∆Θ, where, as customary, ties are broken in favor of the seller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that a buyer’s type k /∈ Rφ,p,π is among those who decide to do not acquire information from the seller, and, thus, they play a best response to the probability distribution µ (instead of a posterior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 7From the point of view of Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2020), this is equivalent to assuming that there is no type reporting stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 8Whenever the expected utility of a buyer’s type is the same by participating in the protocol as not doing that, we assume that they take the option maximizing the seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 6 ARXIV PREPRINT - FEBRUARY 1, 2023 Finally, when dealing with protocols without menus, it will be useful to directly work with distributions over posteriors induced by signaling schemes, rather than with signaling schemes (Kamenica and Gentzkow, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A signaling scheme φ : Θ → ∆S induces a distribution γ over ∆Θ, which has a support supp(γ) := {ξs | s ∈ S} and satisfies the following conditions: � ξ∈supp(γ) γξ ξθ = µθ ∀θ ∈ Θ, (1) where γξ ∈ [0, 1] is the probability that γ assigns to the posterior ξ ∈ supp(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, instead of working with signaling schemes φ, one can w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' work with distributions γ over ∆Θ that are consistent with the probability distribution µ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', they satisfy the condition in Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' When working with distributions over posteriors γ rather than with signaling schemes φ, with a slight abuse of notation, we denote a seller’s protocol without menus as (γ, p, π), by identifying a signaling scheme with its induced distribution over posteriors γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly, we slightly abuse notation in payment functions, by assuming that they are defined over posteriors rather than signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, we let π : ∆Θ × A → R+, with π(ξ, a) denoting how much the buyer pays back the seller when the induced posterior is ξ ∈ ∆Θ and they play action a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 3 Computing a Seller-optimal Protocol with Menus We begin by studying the problem of computing a seller-optimal protocol in which the seller has the ability of propos- ing a menu of signaling schemes and payment functions to the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, the problem of computing an optimal IC and IR protocol with menus can be formulated as follows: sup φk θ(s)≥0 pk≥0 πk(s,a)≥0 � k∈K λk �� s∈S � θ∈Θ µθφk θ(s) � us θ(bk ξs,πk) − πk(s, bk ξs,πk) � + pk � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2a) � s∈S � θ∈Θ µθφk θ(s) � uk θ(bk ξs,πk) + πk(s, bk ξs,πk) � − pk ≥ � s∈S max a∈A � θ∈Θ µθφk′ θ (s) � uk θ(a) + πk′(s, a) � − pk′ ∀k ∈ K, ∀k′ ∈ K (2b) � s∈S � θ∈Θ µθφk θ(s) � uk θ(bk ξs,πk) + πk(s, bk ξs,πk) � − pk ≥ max a∈A � θ∈Θ µθuk θ(a) ∀k ∈ K (2c) � s∈S φk θ(s) = 1 ∀k ∈ K, ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2d) Notice that Problem (2) is defined in terms of sup rather than max since, as it is the case in principal-agent problems (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', (Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2022)), it is not in general immediate to establish whether the seller’s optimization problem always admits a maximum or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, in the following we show that our problem always admits a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a first step, we prove that we can focus w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g on protocols which are direct and persuasive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given any IC and IR seller’s protocol, it is always possible to recover an IC and IR seller’s protocol that is direct and persuasive, and it provides the seller with the same expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, Lemma 1 follows from the fact that, given any signaling scheme φk and price function πk corresponding to some type k ∈ K, if two signals induce the same best response for a buyer of type k, then it is possible to merge the two signals in a single one, recovering a new signaling scheme and a new price function for type k that achieve the same seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By doing such a procedure for every buyer’s type until there are no two signals inducing the same best response for that type, we obtain a protocol that is direct and persuasive, and it has the same seller’s expected utility as the original protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that, since in direct protocols it holds § = A, whenever we write πk(a, a′) for a, a′ ∈ A, the first action a is the seller’s recommendation (signal), while the second action a′ is the one actually played by the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a second crucial step, we exploit Lemma 1 in order to show that, given an IC and IR protocol that is direct and persuasive, there exists another IC and IR protocol which is still direct and persuasive, it achieves the same seller’s expected utility, and it is such that: (i) for every k ∈ K, the price pk of φk is equal to entire budget bk of a buyer of type k, and (ii) the buyer is not paid back (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', they get a null payment) if they deviate from the seller’s action recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: 7 ARXIV PREPRINT - FEBRUARY 1, 2023 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given an IC and IR protocol {(φk, pk, πk)}k∈K that is direct and persuasive, it is always possible to recover an IC and IR protocol {(φk, ˜pk, ˜πk)}k∈K such that: it is direct and persuasive, it provides the same seller’s expected utility as the original protocol, and, for every buyers’ type k ∈ K, it satisfies ˜pk = bk and ˜πk(a, a′) = 0 for all a ̸= a′ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a direct consequence of Lemma 2, we can compactly denote πk(a, a) as πk(a) for every a ∈ A, since we can focus w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' on payment functions such that πk(a, a′) = 0 for all a ̸= a′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We are now ready to introduce an LP with polynomially-many variables and constraints that is a linear relaxation of Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In order to formulate the LP, we exploit Lemmas 1 and 2 to restrict the attention to direct and persuasive protocols, prices such that pk = bk for every k ∈ K, and payments such that πk(a, a′) = 0 for every k ∈ K and a ̸= a′ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, we encode the terms � θ∈Θ µθφk θ(a)πk(a) as single variables lk(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, the LP reads as follows: max φk θ (a)≥0 lk(a)≥0 yk,k′,a≥0 � k∈K λk � a∈A �� θ∈Θ µθφk θ(a)us θ(a) − lk(a) � + bk s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (3a) � a∈A �� θ∈Θ µθφk θ(a)uk θ(a) + lk(a) � − bk ≥ � a∈A yk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='a − bk′ ∀k ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ∀k′ ∈ K (3b) yk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='a ≥ � θ∈Θ µθφk′ θ (a)uk θ(a) + lk′(a) ∀k ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ∀k′ ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ∀a ∈ A (3c) yk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='a ≥ � θ∈Θ µθφk′ θ (a)uk θ(a′) ∀k ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ∀k′ ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ∀a ̸= a′ ∈ A (3d) � a∈A �� θ∈Θ µθφk θ(a)uk θ(a) + lk(a) � − bk ≥ � θ∈Θ µθuk θ(a′) ∀k ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ∀a′ ∈ A (3e) � θ∈Θ µθφk θ(a)uk θ(a) + lk(a) ≥ � θ∈Θ µθφk θ(a)uk θ(a′) ∀k ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ∀a ̸= a′ ∈ A (3f) � a∈A φk θ(a) = 1 ∀k ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (3g) In LP (3), Constraints (3b)–(3d) ensure that the protocol is IC, Constraints (3e) enforce that it is IR, while Con- straints (3f) guarantee that the protocol is persuasive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given how LP (3) is obtained from Problem (2), it is not immediately clear how, given a feasible solution to LP (3), one can recover a protocol that is a solution to Problem (2) with seller’s expected utility equal to the value of the solution to LP (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, in a solution to LP (3), a variable lk(a) could be strictly positive even when the variables φk θ(a) are equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In such a case, it is not possible to immediately recover a value for πk(a) starting from a solution to LP (3), since lk(a) encodes � θ∈Θ µθφk θ(a)πk(a), from which computing πk(a) would require a division by zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In the following, we show how, given an optimal solution to LP (3), it is indeed possible to build in polynomial time a seller-optimal protocol with menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First, we prove a preliminary result: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The optimal value of LP (3) is at least as large as the supremum in Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, we show that, given a solution to LP (3), it is possible to recover in polynomial time an IC and IR protocol with at least the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a feasible solution to LP (3), it is possible to recover in polynomial time an IC and IR protocol whose seller’s expected utility is greater than or equal to the value of the solution to LP (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, Lemma 4 is proved by showing that, given a feasible solution to LP (3), it is possible to efficiently construct a new solution in which, whenever some variable lk(a) > 0, then there exists at least one state of nature θ ∈ Θ for which φk θ(a) > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', action a is recommended with strictly positive probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, such a procedure does not detriment the objective function value and retains the IC and IR conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, from the new solution, one can recover a protocol that is a valid solution to Problem (2), by letting πk(a) = lk(a)/ � θ∈Θ µθφk θ(a) for all k ∈ K and a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 8 ARXIV PREPRINT - FEBRUARY 1, 2023 Finally, by exploiting Lemmas 3 and 4, we can design a polynomial-time algorithm that finds a seller-optimal protocol with menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, the algorithm can simply optimally solve LP (3) (in polynomial time), and use Lemma 4 to recover an IC and IR protocol having at least the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Tanks to Lemma 3, such a protocol is optimal for the seller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' There exists a polynomial-time algorithm that computes a protocol with menus that maximizes the seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 1 also shows as a byproduct that Problem (2) always admits a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let us remark that the idea of formulating a linear relaxation of a quadratic problem by introducing a new variable has already been used in generalized principal-agent problems by Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' However, in such a setting, the linear relaxation cannot be used to solve the principal’s optimization problem exactly, but only to recover a desirable approx- imation of an optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This is because the problem may not admit a maximum, as shown by Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022b) even in the special case of hidden-action principal-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Surprisingly, in our information-selling setting, the linear relaxation can be used to find an (exact) optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, this is possible since, in our setting, the seller observes the action undertaken by the buyer, while in hidden-action principal-agent problems the principal does not directly observe the agent’s action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 4 Drawing a Connection with Principal-agent Problems In this section, we show that our information-selling problem is intimately related to a particular class of principal- agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Specifically, we show that the problem of computing a seller-optimal protocol without menus is a generalization of the problem of computing an optimal contract in principal-agent problems in which the principal observes the action undertaken by the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='1, we formally introduce principal-agent problems with observable actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2, we show how such problems are related to our information-selling problem, and we prove an hardness result for them which carries over to our problem Finally, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='3, we provide some preliminary technical results that will be useful in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='1 Principal-agent Problem with Observable Actions We start by formally defining an instance of (Bayesian) observable-action principal-agent problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='9 For ease of ex- position, we reuse some of the notation already introduced in Section 2, in order to denote elements that in observable- action principal-agent problems have the same role as in our information-selling setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The agent has a finite set K of possible types, and a type k ∈ K is drawn with probability λk according to a known distribution λ ∈ ∆K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Each agent’s type k ∈ K has a set A of actions, with each action having a type-dependent cost ck a ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The principal is characterized by a reward ra ∈ [0, 1] for every agent’s action a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the principal can commit to a contract, which can be encoded by a function π : A → R+ defining a payment π(a) from the principal to the agent for every possible agent’s action a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a contract, an agent of type k ∈ K plays a best response bk π ∈ A, defined as bk π ∈ arg maxa∈A � π(a) − ck a � , where, as usual, we assume that ties are broken in favor of the principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, the principal’s goal is to commit to a contract maximizing their expected utility, which is defined as � k∈K λk[rbkπ −π(bk π)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2 From Selling Information to Observable-action Principal-agent Problems Next, we show that our information-selling problem in the case in which protocols are without menus and the buyer has limited liability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', bk = 0 for all k ∈ K) is strongly related to the problem of finding an optimal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', expected-utility- maximizing) contract in observable-action principal-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Specifically, we show that, given a posterior ξ ∈ ∆Θ, designing a payment function π : ∆Θ × A → R+ that maximizes the seller’s expected utility conditioned on the fact that the induced posterior is ξ is equivalent to finding an optimal contract in a suitably-defined principal-agent problem with observable actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, for ease of presentation, we introduce the following notion of payment function that is optimal for the seller in a given posterior: 9Notice that observable-action principal-agent problems are a special case of Bayesian hidden-action principal-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, this can be easily seen by taking an instance of the hidden-action problem in which outcomes correspond one-to-one with agent’s actions, and each action deterministically determines its corresponding outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 9 ARXIV PREPRINT - FEBRUARY 1, 2023 Definition 3 (Optimal payment function in a posterior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a posterior ξ ∈ ∆Θ, we say that a payment function π : ∆Θ × A → R+ is optimal in ξ if the following holds: π ∈ argmax π′ � k∈K λk �� θ∈Θ ξθ us θ(bk ξ,π′) − π(ξ, bk ξ,π′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (4) Notice that, in Problem (4), the price p of the signaling scheme φ does not appear in the seller’s expected utility, since we are restricted to settings in which the buyer has limited liability, and, thus, it is always the case that p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For the same reason, we can safely assume that all the buyer’s types satisfy IR constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, we can state the following crucial result: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a posterior ξ ∈ ∆Θ, solving Problem (4) is equivalent to computing a contract maximizing the principal’s expected utility in an instance of observable-action principal-agent problem such that, for every agent’s type k ∈ K and action a ∈ A, the following holds: ck a = � θ∈Θ ξθ � uk θ(bk ξ) − uk θ(a) � and ra = � θ∈Θ ξθ us θ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, finding an optimal contract in any instance of observable-action principal-agent problem can be reduced in polynomial time to computing a seller-optimal protocol without menus in a problem instance in which the buyer has limited liability and there is only one state of nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The first statement in Lemma 5 implies that, given an instance of our information-selling problem in which the buyer has limited liability and there is only one state of nature, it is possible to compute a seller-optimal protocol without menus by finding an optimal contract in an instance of observable-action principal-agent problem defined as in the lemma (notice that such an instance can be easily built in polynomial time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, by Lemma 5, we can easily prove the following: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances in which the buyer has limited liability and there is only one state of nature, computing a seller-optimal protocol without menus is equivalent to the problem of finding an optimal contract in general instances of the observable-action principal-agent problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' While the computational complexity of finding optimal contracts in hidden-action principal-agent problems is well understood (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', (Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2022a)), to the best of our knowledge, there are no results on problems with observable actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In following theorem, we prove a strong hardness result for them: there exists a constant α < 1 such that designing a contract which provides the principal with at least an α fraction of the expected utility in an optimal contract is computationally intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In observable-action principal-agent problems, the problem of computing a contract maximizing the principal’s expected utility is APX-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, Theorem 2 immediately gives the following result: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The problem of computing a seller-optimal protocol without menus is APX-hard, even when the buyer has limited liability and the number of states of nature d is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As we show in the following sections (see Theorems 8 and 11), whenever the number of states of nature is fixed, the problem of computing a seller-optimal protocol without menus admits a polynomial-time algorithm providing a particular bi-criteria approximation of the seller’s expected utility in an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Such an approximation is similar to the the bi-criteria guarantees provided by Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022a) for Bayesian hidden-action principal- agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By Theorem 2, our polynomial-time bi-criteria approximation algorithm for the setting in which the buyer has limited liability (Theorem 8) can be easily adapted to work with observable-action principal-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 7 in (Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2022a) shows that, for hidden-action problems, such bi-criteria approximations are tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We leave as an open problem to establish whether these are also tight in our observable-action principal-agent problems or one can obtain better guarantees in polynomial time for our specific case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='3 Additional Preliminary Technical Results We conclude the section by recalling two already-known results on hidden-action principal-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Clearly, these also hold for the specific case of observable-action principal-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By Theorem 2, such results can be easily cast to our information-selling problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, we also show that one of them can be strengthen in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The first result that we are going to introduce makes use of linear contracts, which are payment schemes that pay the agent a given fraction of the principal’s reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, in observable-action principal-agent problems, a contract 10 ARXIV PREPRINT - FEBRUARY 1, 2023 π : A → R+ is said to be linear if there exists a β ∈ [0, 1] such that π(a) = β ra for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Despite their simplicity, linear contracts provide good approximations with respect to general ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In particular, the following holds: Theorem 4 (Essentially Theorem 3 by Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In an observable-action principal-agent problem, for any ρ ∈ (0, 1/2], there exists a linear contract π : A → R+ such that: � k∈K λk � rbkπ − π(bk π) � ≥ ρ max π′ � k∈K λk � rbk π′ − π′(bk π′) � − 2Ω(1/ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, such a linear contract is defined by a parameter β = 1 − 2−i, for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', ⌊1/2ρ⌋}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We will make use of a stronger version of Theorem 4, which applies to our setting and directly follows from the analysis of Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022a) and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a posterior ξ ∈ ∆Θ, for any ρ ∈ (0, 1/2], there exists a payment function π : ∆Θ × A → R+ such that π(ξ, a) = β � θ∈Θ ξθ us θ(a) for every a ∈ A, where β ∈ [0, 1] is an (action-independent) parameter, and, additionally, the following holds: � k∈K λk � θ∈Θ ξθ � us θ(bk ξ,π) − π(s, bk ξ,π) � ≥ ρ � k∈K λk max a∈A � θ∈Θ ξθ � us θ(a) + uk θ(a) − uk θ(bk ξ) � − 2Ω(1/ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, such a parameter β is equal to 1 − 2−i for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' , ⌊1/2ρ⌋}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, we recall a useful result that establishes a connection between agent’s best responses and approximate best responses in principal-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, such a result states that, given a contract under which the agent is allowed to play an ǫ-best response (for some ǫ ≥ 0), it is always possible to recover a new contract in which the agent must play an (exact) best response, by only incurring in a small loss in the principal’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, given ǫ ≥ 0 and a contract π : A → R+, for every k ∈ K, we let Bk,ǫ π ⊆ A be the set of ǫ-best-response actions for an agent of type k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Such a set is made by all the actions a ∈ A such that π(a) − ck a ≥ maxa′∈A � π(a′) − ck a′ � − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We denote by bk,ǫ π ∈ Bk,ǫ π an ǫ-best-response action that is actually played by an agent of type k, assuming that ties are broken in favor of the principal, as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then: Theorem 5 (Essentially Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='4 by Dutting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given ǫ ≥ 0, an instance of observable-action principal-agent problem and, and a contract π : A → R+, there exists a contract π′ : A → R+ such that π′(a) = (1 − √ǫ) π(a) + √ǫ ra for every a ∈ A, and the following holds: � k∈K λk � rbk π′ − π′(bk π′) � ≥ � k∈K λk � rbk,ǫ π − π(bk,ǫ π ) � − 2√ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 5 can be easily cast to our setting by means of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given ǫ ≥ 0, a posterior ξ ∈ ∆Θ, and a payment a function π : ∆Θ × A → R+, there exists a payment function π′ : ∆Θ × A → R+ such that π′(ξ, a) = (1 − √ǫ) π(ξ, a) + √ǫ � θ∈Θ ξθus θ(a) for every a ∈ A, and the following holds: � k∈K λk �� θ∈Θ ξθ us θ(bk ξ,π′) − π′(ξ, bk ξ,π′) � ≥ � k∈K λk �� θ∈Θ ξθ us θ(bk,ǫ ξ,π) − π(ξ, bk,ǫ ξ,π) � − 2√ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Corollary 3 will be crucial to provide our results in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 5 Computing a Seller-optimal Protocol without Menus: The Case of a Buyer with Limited Liability In this section, we study the problem of computing a seller-optimal protocol without menus when the buyer has limited liability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', each buyer’s types k ∈ K has budget bk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As remarked in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2, such a setting is of interest on its own, since it is a generalization of the one addressed by Dughmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the technical results derived in this section will be useful to deal with the general problem in which the buyer has no limited liability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show how to circumvent the APX-hardness result that we established in Corollary 1, first, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='1, by fixing the number of buyer’s actions, and then, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2, by fixing the number of states of nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In this section, since the buyer has limited liability, we can assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' that p = 0, so that we can compactly denote a protocol with a pair (γ, π), rather than with (γ, p, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 11 ARXIV PREPRINT - FEBRUARY 1, 2023 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='1 Fixing the Number of Buyer’s Actions First, we show that, whenever the buyer has limited liability and the number of buyer’s actions m is fixed, the problem of computing a seller-optimal protocol without menus admits a PTAS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', we can design a protocol whose seller’s expected utility is arbitrarily close to that of an optimal protocol in time polynomial in the instance size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In order to design our PTAS, we start by observing that, since p = 0, the IR constraints are satisfied by all the protocols (γ, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This allows us to formulate the problem of computing a seller-optimal protocol without menus as the following optimization problem:10 max γξ≥0 π(ξ,a)≥0 � k∈K λk � ξ∈supp(γ) γξ �� θ∈Θ ξθ us θ(bk ξ,π) − π(ξ, bk ξ,π) � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (5a) � ξ∈supp(γ) γξ ξθ = µθ ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (5b) Notice that Problem (5) is defined over general distributions over posteriors γ, whose support supp(γ) may be not finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, as we show in the following, the crucial result that we need to design a PTAS is the possibility of restricting the attention to finite sets of posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We need to introduce a particular class of posteriors, which are called q-uniform posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Definition 4 (q-Uniform posterior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A posterior ξ ∈ ∆Θ is q-uniform if it can be obtained by averaging the elements of a multi-set defined by q ∈ N>0 canonical basis vectors of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In the following, we denote by Ξq ⊆ ∆Θ (for a given q ∈ N>0) the finite set of all the q-uniform posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As it is easy to check, such a set satisfies |Ξq| ≤ min{dq, qd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In order to derive our PTAS, as a first preliminary result we show that, given any posterior ξ∗ ∈ ∆Θ, payment function π : ∆Θ × A → R+, and ǫ > 0, there always exists a signaling scheme γ supported on Ξq which induces posterior ξ∗ on average and guarantees a seller’s expected utility close to that provided by the posterior ξ∗ (assuming the buyer plays an ǫ-best response).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given any ǫ, α > 0, a posterior ξ∗ ∈ ∆Θ, and a payment function π : ∆Θ × A → R+, there always exists a signaling scheme γ ∈ ∆Ξq with q = 2 log(2m/α)/ǫ2 such that: � ξ∈Ξq γξ �� θ∈Θ ξθ us θ(bk,ǫ ξ,π) − π(ξ, bk,ǫ ξ,π) � ≥ � θ∈Θ ξ∗ θ us θ(bk ξ∗,π) − π(ξ∗, bk ξ∗,π) − α, for every buyer’s type k ∈ K, where we let π : ∆Θ ×A → R+ be a payment function that is optimal in every posterior ξ ∈ Ξq when the buyer plays an ǫ-best response, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', π solves Problem (4) for every ξ ∈ Ξq with bk ξ,π replaced by bk,ǫ ξ,π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Furthermore, the signaling scheme γ satisfies: � ξ∈Ξq γξ ξθ = ξ∗ θ ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Lemma 6 guarantees that, by decomposing each posterior ξ ∈ ∆Θ as a convex combination of the elements of Ξq, the seller’s expected utility decreases by at most α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This implies that, assuming the buyer plays an ǫ-best response, it is possible to work with signaling schemes (and thus payment functions) supported on Ξq, by only slightly degrading the seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Another component that we need for our PTAS is an algorithm that, given a q-uniform posterior, computes an optimal payment function in that posterior (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', a payment function solving Problem (4) for such a posterior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By Theorem 2, it is easy to see that such an algorithm has to solve a problem that is equivalent to computing an optimal contract in observable-action principal-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, by Theorem 3, such a problem is APX-hard in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Next, we show that, whenever the number of buyer’s actions m is fixed, the APX-hardness result can be circumvented, and, thus, we can provide an algorithm that solves the desired task and runs in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: 10Notice that, as it is the case for Problem (2) in Section 3, it is not immediately clear a priori whether the problem of computing a seller-optimal protocol without menus admits a maximum or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, in principle we should start by defining the problem with a sup rather than a max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' However, in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2, we provide a (possibly exponential-time) algorithm which finds a seller-optimal protocol without menus in general settings, and this implies that a maximum always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 12 ARXIV PREPRINT - FEBRUARY 1, 2023 Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances where the buyer has limited liability and the number of buyer’s actions m is fixed, there exists a polynomial-time algorithm that, given a posterior ξ ∈ ∆Θ as input, computes the payments π(ξ, a) for a ∈ A of a payment function π : ∆Θ × A → R+ optimal in ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that, in order to get a payment function π : ∆Θ × A → R+ that is optimal in every posterior ξ ∈ Ξq, it is sufficient to apply Lemma 7 for each ξ ∈ Ξq, then putting together all the computed payments π(ξ, a) in order to obtain the overall payment function π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The final piece that we need to complete the design of our PTAS is a way of coming back to work with buyer’s best responses, rather than using ǫ-best responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, this is possible thanks to Corollary 3, which allows us to modify the payment function in all the induced posteriors, so as to achieve the desired result by only losing a small amount 2√ǫ of the seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Now, we are ready to design our PTAS that works whenever the buyer has limited liability and the number of buyer’s actions m is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By Lemma 6, we can focus on signaling schemes supported over q-uniform posteriors, for a suitably-defined q ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, thanks to Lemma 7, we can compute a payment function that is optimal in all the q-uniform posteriors, by running the polynomial-time algorithm in Lemma 7 for each q-uniform posterior in Ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By Corollary 3, such an optimal payment function achieves a seller’s expected utility that is close to that obtained by a payment function which is optimal in every q-uniform posterior when considering ǫ-best responses, thus allowing for the application of the result in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In conclusion, our PTAS works by solving a modified version of LP (5), where we set supp(γ) := Ξq in Equation (5a), and we take as payment function the one returned by applying Lemma 7 in each posterior ξ ∈ Ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It is easy to see that the overall procedure requires time polynomial in the instance size when the number of actions m is fixed, since |Ξq| ≤ dq and q = 2 log(2m/α)/ǫ2 as prescribed by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' However, the overall running time depends exponentially in α > 0, which the seller’s expected utility approximation provided by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This allows us to prove the following result: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances where the buyer has limited liability and the number of buyer’s actions m is fixed, the problem of computing a seller-optimal protocol without menus admits a PTAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, we show that a similar approach can be employed to derive a quasi-polynomial time algorithm providing a bi-criteria approximation of the seller’s expected utility in an optimal protocol, even when the number of buyer’s actions m is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, in our PTAS, the computation of an optimal payment function in a given q-uniform posterior can be done in polynomial time only when the number of actions is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' While in general the problem is APX-hard, an approximately-optimal price function can be computed in polynomial time by applying Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, since q = 2 log(2m/α)/ǫ2 and |Ξq| ≤ dq, the enumeration over the q-uniform posteriors can be performed in time quasi polynomial in th number of actions m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This gives the following result: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances in which the buyer has limited liability, there exists an algorithm that, given any α, ǫ > 0 and ρ ∈ (0, 1/2] as input, returns a protocol without menus achieving a seller’s expected utility greater than or equal to ρ OPT − 2−Ω(1/ρ) − (α + 2√ǫ), where OPT is the seller’s expected utility in an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the algorithm runs in time polynomial in Ilog m—where I is the size of the problem instance and m is the number of buyer’s actions—, and the seller’s expected utility in the returned protocol is greater than or equal to OPTLIN − (α + 2√ǫ), where OPTLIN is the best expected utility achieved by a protocol parametrized by β as in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The second part of the statement will be useful in deriving our results for the problem of computing seller-optimal protocols without menus in the general case in which the buyer has no limited liability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, it states that, even if our approximation algorithm only provides a bi-criteria approximation of a seller-optimal protocol, the returned protocol achieves a seller’s expected utility which is arbitrarily close to that achievable by using payment functions that define the payments as a given fraction of the seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2 Fixing the Number of States of Nature Next, we study the case in which the buyer has limited liability and the number of states of nature d is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We prove that, in such a setting, it is possible to compute a bi-criteria approximation of an optimal protocol without menus similar to that in Theorem 7, but in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that such a result circumvents the APX-hardness one provided in Corollary 1, as the latter is based on a reduction working with instances with only one state of nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='1, we first show that it is possible to employ signaling schemes supported on the set Ξq of q-uniform posteriors (for a suitably-defined q ∈ N>0), by only suffering an arbitrarily small, additive loss in terms of seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' While the following result is similar to the one obtained in Lemma 6, it is based on different techniques and, in particular, on the fact that the seller’s expected utility is Lipschitz continuous in the buyers’ posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: 13 ARXIV PREPRINT - FEBRUARY 1, 2023 Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given any α > 0, a posterior ξ∗ ∈ ∆Θ, and a payment function π : ∆Θ × A → R+ that is optimal in every posterior ξ ∈ Ξq with q = ⌈9d/α2⌉, there exists a signaling scheme γ ∈ ∆Ξq: � ξ∈Ξq γξ �� θ∈Θ ξθ us θ(bk ξ,π) − π(ξ, bk ξ,π) � ≥ � θ∈Θ ξ∗ θus θ(bk ξ∗,π) − π(ξ∗, bk ξ∗,π) − α, for every receiver’s type k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Furthermore, the signaling scheme γ satisfies: � ξ∈Ξq γξ ξθ = ξ∗ θ ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly to the case of a fixed number of actions, we employ Lemma 8 to restrict the attention to signaling schemes (and thus payment functions) supported on Ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, in this case, we can apply Corollary 2 in each q-uniform posterior in order to compute in polynomial time a payment function that provides a bi-criteria approximation of the optimal seller’s expected utility in such a posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, we design an algorithm that solves a modified version of LP (5), where we set supp(γ) := Ξq in Equation (5a), and we take as payment function the one obtained by putting together those computed by means of Corollary 2 for each ξ ∈ Ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, the overall procedure requires polynomial time, since |Ξq| ≤ qd and the number of states of nature d is fixed, and achieves a bi-criteria approximation of the seller’s expected utility in an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances in which the buyer has limited liability and the number of states of nature d is fixed, there exists an algorithm that, given α > 0 and ρ ∈ (0, 1/2] as input, returns in polynomial time a protocol without menus achieving a seller’s expected utility greater than or equal to ρ OPT − 2−Ω(1/ρ) − α, where OPT is the seller’s expected utility in an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the seller’s expected utility in the returned protocol is greater than or equal to ρ OPTLIN − 2−Ω(1/ρ) − α where OPTLIN is the best expected utility achieved by a protocol parametrized by β as in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly to Theorem 7, the second part of the statement will be useful for deriving our results in the general case in which the buyer has no limited liability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 6 Computing a Seller-optimal Protocol without Menus: The General Case We conclude our analysis by considering the problem of computing a seller-optimal protocol without menus in general instances in which the buyer has no limited liability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, in such a setting, the seller also decides a price p ∈ R+ for the signaling scheme proposed to the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First, we provide a negative result for general instances that is stronger than the one established in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In particular, the latter result states that the seller’s optimization problem is APX-hard even in the special case in which the buyer has limited liability and there is only one state of nature, relying on a reduction employing instances with an arbitrary number of actions m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, for the specific case in which the buyer has limited liability and the number of actions m is fixed, Theorem 6 provides a PTAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Next, we show that, in general instances where the buyer may not have limited liability, the problem is APX-hard even when the number of buyer’s actions m is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' To prove such an hardness result, we employ a result by Guruswami and Raghavendra (2009) (see Theorem 9 below), which is about the following promise problem related to the satisfiability of a fraction of linear equations with rational coefficients and variables restricted to the hypercube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='11 Definition 5 (LINEQ-MA(1−ζ, δ) by Guruswami and Raghavendra (2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For any two constants ζ, δ ∈ R satisfying 0 ≤ δ ≤ 1 − ζ ≤ 1, LINEQ-MA(1 − ζ, δ) is the following promise problem: Given a set of linear equations Ax = c over variables x ∈ Qnvar, with coefficients A ∈ Qneq×nvar and c ∈ Qneq, distinguish between the following two cases: there exists a vector ˆx ∈ {0, 1}nvar that satisfies at least a fraction 1 − ζ of the equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' every possible vector x ∈ Qnvar satisfies less than a fraction δ of the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 9 (Guruswami and Raghavendra (2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For all the constants ζ, δ ∈ R which satisfy 0 ≤ δ ≤ 1 − ζ ≤ 1, the problem LINEQ-MA(1 − ζ, δ) is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 11In the definition in (Guruswami and Raghavendra, 2009), the vector ˆx can be non-binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' How- ever, Guruswami and Raghavendra (2009) use a binary vector ˆx in their proof and, thus, their hardness result also holds for our definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 14 ARXIV PREPRINT - FEBRUARY 1, 2023 Then, Theorem 5 allows us to prove the following hardness result: Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The problem of computing a seller-optimal protocol without menus is APX-hard, even when the number of buyer’s actions m is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In the following, we show how to circumvent the hardness result in Theorem 10, by providing, in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='1, a quasi-polynomial-time bi-criteria approximation algorithm and, in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2, a polynomial-time (exact) algorithm working when the number of buyer’s types is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='1 A General Quasi-polynomial-time Bi-criteria Approximation Algorithm In order to circumvent the negative result presented in Theorem 10, we design a quasi-polynomial-time algorithm that computes a protocol without menus providing a bi-criteria approximation of the seller’s expected utility in an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, our algorithm guarantees a multiplicative approximation ρ of the optimal utility, by only suffering an additional 2−Ω(1/ρ) + α additive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, we show that our algorithm runs in polynomial time whenever either the number of buyer’s actions m or that of states of nature d is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In order to prove the approximation guarantees of our algorithm, we rely on Theorems 7 and 8, and we decompose the seller’s expected utility in an optimal protocol without menus into the sum of three different terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Our algorithm works by computing three protocols without menus, each one approximating one of the three terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Choosing the best protocol among the three provides the desired approximation guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The following is an intuition of how each term composing the optimal seller’s expected utility is approximated by our algorithm: The first term is related to the seller’s expected utility collected from buyer’s types for which the IR constraints are not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Such a utility term can be trivially achieved by a protocol that charges no price, discloses no information, and never pays back the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The second term is related to the best seller’s expected utility which can be extracted from a buyer’s action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This is related to the optimal seller’s expected utility in a setting with limited liability, since, in that case, the seller’s expected utility is determined by the buyer’s action only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, the second utility term can be approximated by using either the algorithm provided in Theorem 7 or that given in Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='12 The third term is related to the seller’s expected utility obtained by the transfers between the seller and the buyer, which include the charged price and the final payment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Such a utility term can be approximated by using a protocol that reveals all the information to the buyer while charging a carefully-chosen price for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, we prove the following main result: Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' There exists an algorithm that, given any α > 0 and ρ ∈ (0, 1/6] as input, computes a protocol without menus whose seller’s expected utility is greater than or equal to ρ OPT − 2−Ω(1/ρ) − α, where OPT is the seller’s expected utility in an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the algorithm runs in time polynomial in Ilog m—where I is the size of the problem instance—when it is implemented with the algorithm in Theorem 7 as a subroutine, while it runs in time polynomial in Id when it is implemented with the algorithm in Theorem 8 as a subroutine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2 Fixing the Number of Buyer’s Types Next, we study the problem of computing a seller-optimal protocol without menus when the number of buyer’s types is fixed, showing that it is possible to design a polynomial-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a byproduct, the existence of such an algorithm shows that, for protocols without menus, the seller’s optimization problem always admits a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a preliminary result, we show that it is always possible to focus on protocols without menus (φ, p, π) that employ signals belonging to the set An, and define signaling schemes φ : Θ → ∆An and payment functions π : An × A → R+ such that, for every signal a ∈ An and k ∈ K, it holds ak ∈ Bk ξa,π, where ak ∈ A denotes the action corresponding to type k in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, in such protocols, a signal specifies an action recommendation for each buyer’s type, so that the buyer is always incentivized to follow such recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' With a slight abuse of notation, we say that protocols without menus (φ, p, π) as described above are generalized-direct and generalized-persuasive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, we prove the following result: Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a seller’s protocol without menus, there always exists another protocol without menus which is generalized-direct and generalized-persuasive, and achieves the same seller’s expected utility as the original protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 12Notice that we cannot employ Theorem 6 in place of Theorem 7, since the latter guarantees to achieve a seller’s expected utility that is arbitrarily close to that of the best protocol employing payment functions parametrized by β, and this is needed in order to derive the guarantees of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Such a guarantee is not provided by Theorem 6, which only predicates on the quality of the returned protocol with respect to an optimal protocol without menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 15 ARXIV PREPRINT - FEBRUARY 1, 2023 In order to prove the lemma, we observe that, given a protocol, if two signals induce the same best response for every buyer’s type, it is always possible to merge the two signals, retaining the same expected utility for both the seller and the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, by iterating such a process, we recover a signaling scheme and a payment function employing An as set of signals .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a second crucial step, we show that we can focus on protocols without menus (φ, p, π) whose price p is equal to the budget bk of one buyer’s type k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a protocol without menus, there always exists another protocol (φ, p, π) such that p = bk for some k ∈ K, while achieving the same seller’s expected utility as the original protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, equipped with Lemma 9 and Lemma 10, we are ready to provide our polynomial-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, since we can restrict the attention to protocols without menus (φ, p, π) that are generalized-direct and generalized- persuasive, and whose prices p belong to the set {bk}k∈K, we can solve the seller’s problem by iterating over all the possible price values p ∈ {bk}k∈K and, for each of them, over all the possible subsets R ⊆ K ∩ {k ∈ K : bk ≥ p} of buyer’s types that satisfy the IR constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This can be done in polynomial time since the number of buyer’s types n is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, for every price value p ∈ {bk}k∈K and set R ⊆ K ∩ {k ∈ K : bk ≥ p}, it is sufficient to solve the following optimization problem: sup φ≥0 π≥0 � k∈R λk � a∈An � θ∈Θ µθφθ(a) [us θ(ak) − π(a, ak)] + � k /∈R λk � θ∈Θ µθus θ(bk µ) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (6a) � θ∈Θ µθφθ(a) � uk θ(ak) + π(a, ak) � ≥ � θ∈Θ µθφθ(a) � uk θ(a′) + π(a, a′) � ∀k ∈ R, ∀a ∈ An, ∀a′ ̸= ak ∈ A (6b) � a∈An � θ∈Θ µθφθ(a) � uk θ(ak) + π(a, ak) � − bk ≥ � θ∈Θ µθuk θ(bk µ) ∀k ∈ R (6c) � a∈An � θ∈Θ µθφθ(a) � uk θ(ak) + π(a, ak) � − bk ≤ � θ∈Θ µθuk θ(bk µ) ∀k ̸∈ R (6d) � a∈An φθ(a) = 1 ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (6e) By using techniques similar to those used in Section 3 for protocols with menus, we can show that Problem (6) is solvable in polynomial time by means of a suitable-defined LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This allows us to state our last results: Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances in which the number of buyer’s types n is fixed, the problem of computing a seller-optimal protocol without menus admits a polynomial-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The problem of computing a seller-optimal protocol without menus always admits a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' References Paola Alimonti and Viggo Kann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Some APX-completeness results for cubic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theoretical 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Bayesian persuasion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' American Economic Review 101, 6 (2011), 2590–2615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Shuze Liu, Weiran Shen, and Haifeng Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Optimal Pricing of Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proceedings of the 22nd ACM Conference on Economics and Computation (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Yoav Shoham and Kevin Leyton-Brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Multiagent systems: Algorithmic, game-theoretic, and logical founda- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 17 ARXIV PREPRINT - FEBRUARY 1, 2023 A Proofs Omitted from Section 3 Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given any IC and IR seller’s protocol, it is always possible to recover an IC and IR seller’s protocol that is direct and persuasive, and it provides the seller with the same expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let { � φk, pk, πk � }k∈K be an IC and IR protocol such that there exist two signals s1, s2 ∈ S inducing the same best response ¯a ∈ A for a given receiver’s type ¯k ∈ K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', b¯k ξs1 = b¯k ξs2 = ¯a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In the following, we show how to replace φ¯k and π¯k with a new signaling scheme ¯φ¯k and a new payment function ¯π¯k by merging s1, s2 into a single signal ¯s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally we define: ¯φ¯k θ(¯s) = φk θ(s1) + φk θ(s2) for each θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly, we define: ¯π ¯k(¯s, ¯a) = � θ∈Θ µθφ¯k θ(s1)π¯k(s1, ¯a) + � θ∈Θ µθφ¯k θ(s2)π¯k(s2, ¯a) � θ∈Θ µθ(φ¯k θ(s1) + φ¯k θ(s2)) , Finally, we does not change all the other components of the protocol i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', we leave these components of { �¯φk, ¯pk, ¯πk � }k∈K equal to the one in { � φk, pk, πk � }k∈K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' To prove the lemma we show that the protocol { �¯φk, ¯pk, ¯πk � }k∈K achieves the same seller’s expected utility of { � φk, pk, πk � }k∈K, while satisfying the IC and IR constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a first step, we observe that: � s∈S\\{s1,s2} � θ∈Θ µθ ¯φ ¯k θ(s) � u ¯k θ(b ¯k ξs,¯π) + ¯π¯k(s, b ¯k ξs,¯π) � + � θ∈Θ µθ ¯φ ¯k θ(¯s)[u ¯k θ(¯a) + ¯π¯k(¯s, ¯a)] − ¯p¯k = � s∈S\\{s1,s2} � θ∈Θ µθφ ¯k θ(s) � u ¯k θ(b ¯k ξs,π) + π¯k(s, b ¯k ξs,π) � + � θ∈Θ µθφ ¯k θ(s1)[u ¯k θ(¯a) + π¯k(s1, ¯a)]+ + � θ∈Θ µθφ ¯k θ(s2)[u ¯k θ(¯a) + π¯k(s2, ¯a)] − p¯k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The latter equality holds by linearity and proves that the protocol {(¯φk, ¯pk, ¯πk)}k∈K preserves the left-hand sides of the IR and IC constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, thanks to the convexity of the max operator, we can show that: max a∈A � θ∈Θ µθφ ¯k θ(s1)[u ¯k θ(a) + π¯k(s1, a)] + max a∈A � θ∈Θ µθφ ¯k θ(s2)[u ¯k θ(a) + π¯k(s2, a)] − p¯k ≥ max a∈A � θ∈Θ µθ ¯φ ¯k θ(¯s)[u ¯k θ(a) + ¯π¯k(¯s, a)] − p¯k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, by summing over the set (S ∪{¯s})\\{s1, s2}, we notice that the value of the right-hand side of the IC constraints achieved by the protocol {(¯φk, ¯pk, ¯πk)}k∈K is less or equal to the the value achieved by {(φk, pk, πk)}k∈K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Due to that, we can easily conclude that the new protocol preserves the IC and the IR constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, by observing that the following equality holds: � θ∈Θ µθ ¯φk θ(¯s)[u ¯k θ(¯a) + ¯π¯k(¯s, ¯a)] = � θ∈Θ µθφ ¯k θ(s1)[u ¯k θ(¯a) + π¯k(s1, ¯a)] + � θ∈Θ µθφ ¯k θ(s2)[u ¯k θ(¯a) + π¯k(s2, ¯a)], we can easily prove that the two protocols achieve the same seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, by iterating this procedure for each buyer’s type and couple of signals until there are no two signals inducing the same best response for that type, we get a protocol that employs direct and persuasive signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given an IC and IR protocol {(φk, pk, πk)}k∈K that is direct and persuasive, it is always possible to recover an IC and IR protocol {(φk, ˜pk, ˜πk)}k∈K such that: it is direct and persuasive, it provides the same seller’s expected utility as the original protocol, and, for every buyers’ type k ∈ K, it satisfies ˜pk = bk and ˜πk(a, a′) = 0 for all a ̸= a′ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We first prove that by setting ˜πk(a, a′) = 0 for each a ̸= a′ ∈ A and k ∈ K, the seller’s expected utility does not change and the IC and IR constraints are preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First, we show that by taking ˜πk(a, a′) = 0 for each a ̸= a′ ∈ A and k ∈ K, the signaling schemes φk remain persuasive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, we have that: � θ∈Θ µθφk θ(a)uk θ(a) + πk(a, a) ≥ � θ∈Θ µθφk θ(a′)uk θ(a′) + πk(a, a′) ≥ � θ∈Θ µθφk θ(a′)uk θ(a′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 18 ARXIV PREPRINT - FEBRUARY 1, 2023 Moreover, the left-hand side of all the equations defining the IR and IC constraints does not change since it depends only from the payments ˜πk(a, a) = πk(a, a) for each a′ ∈ A that remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Furthermore, by setting ˜πk(a, a′) = 0 for each a ̸= a′ ∈ A and k ∈ K, the right-hand sides of the IC constraints achieve smaller or equal values, since, intuitively, we are setting to zero non-negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the IC constraints are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, by observing that the left-hand sides of the IR constraints and the seller’s expected utility do not embed terms ˜πk(a, a′) with a ̸= a′ ∈ A and k ∈ K, we conclude the first part of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In the second part of the proof we show that it is always possible to define a protocol in which the seller asks to each buyer’s type to deposit all their budget at the beginning of the interaction, achieving the same seller’s ex- pected utility and satisfying the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, we show that given a protocol { � φk, pk, πk � }k∈K the protocol { � φk, ˜pk, ˜πk � }k∈K, with ˜pk = bk and ˜πk(a) = πk(a, a) + bk − pk for each k ∈ K and a ∈ A, achieves the same seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed by linearity we have: � k∈K λk � � a∈A � θ∈Θ µθφk θ(a)us θ(a) − πk(a) + pk � = � k∈K λk � � a∈A � θ∈Θ µθφk θ(s)us θ(a) − πk(a) + bk − bk + pk � = � k∈K λk � � a∈A � θ∈Θ µθφk θ(a)us θ(a) − ˜πk(a) + bk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' With similar arguments it is easy to check that the protocol { � φk, ˜pk, ˜πk � }k∈K satisfies the IC and IR constraints, concluding the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The optimal value of LP (3) is at least as large as the supremum in Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that for each protocol { � φk, pk, πk � }k∈K that is feasible for Problem (2), we can derive a solution to LP (3) with at least the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This is sufficient to prove the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By Lemma 1 and Lemma 2, we focus without loss of generality on protocols { � φk, pk, πk � }k∈K that are direct and persuasive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, we can build a solution (¯φ, ¯l, ¯y) to LP (3) letting ¯lk(a) = � θ∈Θ µθφθ(a)πk(a) for each k ∈ K and a ∈ A, and ¯yk,k′,a = max �� θ∈Θ µθφk′ θ (a) � uk θ(a) + πk′(a, a) � , max a′̸=a � θ∈Θ µθφk′ θ (a′)uk θ(a′) � for each k, k′ ∈ K and a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, we let ¯φ = φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It is easy to verify that the solution (¯φ, ¯l, ¯y) results feasible for LP (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a feasible solution to LP (3), it is possible to recover in polynomial time an IC and IR protocol whose seller’s expected utility is greater than or equal to the value of the solution to LP (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a first step we show that from a feasible solution (φ, l, y) to LP (3) we can recover another solution with at least the same value in which if πk(a) > 0 then there exists a θ ∈ Θ such that φk θ(a) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Specifically, given a k ∈ K and an a ∈ A such that φk θ(a) = 0 for all θ ∈ Θ and lk(a) > 0, let ¯a ∈ A be (¯a, ¯θ) be any couple of an action and a state such that φk ¯θ(¯a) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We now define a new feasible solution (¯φ, ¯l, ¯y) as follows: ¯lk(a) = 0 ¯lk(¯a) = lk(a) + lk(¯a) ¯yk′,k,a = 0 ∀k′ ∈ K ¯yk′,k,¯a = yk′,k,¯a + lk(a) ∀k′ ∈ K, while we leave all the other terms variables equal to the ones in (φ, l, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It is easy to check that the solution (¯φ, ¯l, ¯y)achieves the same seller’s expected utility while satisfying the constrains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Applying this procedure for each couple (k, a) such that φk θ(a) = 0 for all θ ∈ Θ and lk(a) > 0, we obtain a new solution (˜φ, ˜l, ˜y) such that if ˜lk(a) > 0 then there exists a θ ∈ Θ such that ˜φk θ(a) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 19 ARXIV PREPRINT - FEBRUARY 1, 2023 To recover a feasible protocol we just need to set payments as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each couple (k, a), if there exists a θ ∈ Θ such that ˜φk θ(a) > 0, we set ˜πk(a) = ˜lk(a)/(� θ∈Θ µθ ˜φk θ(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Otherwise, we set ˜πk(a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that the ratio ˜lk(a)/(� θ∈Θ µθ ˜φk θ(a)) is always well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, it is easy to see that {˜φk, ˜pk, ˜πk}k∈K, with ˜pk = bk for each k ∈ K is a feasible solution to Problem (2) with at least the same value as (φ, l, y) for LP (3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' There exists a polynomial-time algorithm that computes a protocol with menus that maximizes the seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The algorithm solves LP 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By Lemma 3, this solution has value greater or equal to the supremum of Program 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, exploiting Lemma 4, we can recover in polynomial-time a protocol with at least the same utility, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', an optimal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' B Proofs Omitted from Section 4 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a posterior ξ ∈ ∆Θ, solving Problem (4) is equivalent to computing a contract maximizing the principal’s expected utility in an instance of observable-action principal-agent problem such that, for every agent’s type k ∈ K and action a ∈ A, the following holds: ck a = � θ∈Θ ξθ � uk θ(bk ξ) − uk θ(a) � and ra = � θ∈Θ ξθ us θ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, finding an optimal contract in any instance of observable-action principal-agent problem can be reduced in polynomial time to computing a seller-optimal protocol without menus in a problem instance in which the buyer has limited liability and there is only one state of nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We start proving the first part of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given an instance of Problem (4), we build an instance of the observable-action principal-agent problem with ck a = � θ∈Θ ξθ � uk θ(bk ξ) − uk θ(a) � and ra = � θ∈Θ ξθ us θ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' To prove the equivalence between the two settings, we first show that the set of best-responses for the two problems coincides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, given a payment function π and a type k ∈ K, let a ∈ Bk ξ,π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, for each a′ ∈ A π(a) − ck a = π(a) − � θ∈Θ ξθ � uk θ(bk ξ) − uk θ(a) � = π(a) + � θ∈Θ ξθuk θ(a) − � θ∈Θ ξθuk θ(bk ξ) ≥ π(a′) + � θ∈Θ ξθuk θ(a′) − � θ∈Θ ξθuk θ(bk ξ) = π(a′) − ck a′, showing that a ∈ Bk π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly, we can prove that if a ∈ Bk π, then a ∈ Bk ξ,π This implies that the set of best responses are equivalent for each payment function π, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', Bk π = Bk ξ,π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, argmax π � k∈K λk � rbkπ − π(bk π) � = argmax π � k∈K λk �� θ∈Θ ξθ us θ(bk π) − � θ∈Θ ξθ � uk θ(bk ξ) − uk θ(bk π) � � = argmax π � k∈K λk �� θ∈Θ ξθ us θ(bk π) + � θ∈Θ ξθuk θ(bk π) � = argmax π � k∈K λk �� θ∈Θ ξθ us θ(bk ξ,π) + � θ∈Θ ξθuk θ(bk ξ,π) � , showing the equivalence between the two problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This proves the first part of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We now show that from an instance of the observable-action principal-agent problem we can always build an instance of the selling-information problem without menus and with only a single state of nature θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In particular, we set us θ(a) = ra for each a ∈ A, and uk θ(a) = 1 − ck a for each a ∈ A and k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Following a analysis similar to the first part of the proof, we can show that the two problems are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 20 ARXIV PREPRINT - FEBRUARY 1, 2023 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In observable-action principal-agent problems, the problem of computing a contract maximizing the principal’s expected utility is APX-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We reduce from vertex cover in cubic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, it is NP-hard to approximate the size of the minimum vertex cover in cubic graphs with an approximation (1 + ε), for a given constant ε > 0 Alimonti and Kann (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let η = ε/7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that an (1 − η)-approximation to the principal-agent problem with observable actions can be used to provide a (1 + ε) approximation to vertex cover, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Consider an instance of vertex cover (V, E) with nodes V and edges E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let ρ = |V | and ℓ = |E|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a vertex v ∈ V , we let E(v) be the set of edges e such that v is one of the extreme of the edge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly, given an edge e ∈ E, let V (e) be the set of vertexes v such that e is an edge with extreme v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We build an instance of the principal-agent problem with observable actions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each vertex v ∈ V , there exists an agent’s type kv, while for each e ∈ E, there exists a type ke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each vertex v ∈ V , there exists an action av and an additional action a−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The cost of a type ke, e ∈ E, is cke a− = 0, cke av = 1 2 if e ∈ E(v) and 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The cost of a type kv, v ∈ V , is ckv av = 0, and 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, the principal’s utility is equal to 1 if the action is in {av}v∈V , while is equal to 0 otherwise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', us av = 1 for each v ∈ V and us a− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' All the types are equally probable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', λk = 1 ρ+ℓ for each k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First, we show that if there exists a vertex cover V ⋆ of size ν, the value of the problem is at least (ρ−ν)+ ν 2 + ℓ 2 ρ+ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Consider the payment function such that π(av) = 1 2 if v ∈ V ⋆ and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A type kv with v /∈ V ⋆ plays the action av and receives a payment of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A type kv with v ∈ V ⋆ plays the action av and receives a payment of 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A type ke plays an action av such that e ∈ E(v) (this action exists by construction) and receives a payment of 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It is easy to see that the expected seller’s utility is (ρ−ν)+ ν 2 + ℓ 2 ρ+ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Suppose that there exists an algorithm that provides a 1 − η approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This implies that the algorithm returns a solution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', a payment function π, with value at least (1 − η) ρ− k 2 + ℓ 2 ρ+ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show how to exploit the payment function π to build a vertex cover of size at most (1 + ε)ν in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In particular, given π we recover a vertex cover ¯V of the desired size as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First, it is easy to see that we can set the payment π(a−) = 0 and payments π(av) ∈ {0, 1 2} for each v ∈ V without decreasing the utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, payments are useful only to change the best response of a type ke, e ∈ E, from a− to av, v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' To do so, it is sufficient a payment of 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, let ¯E be the set of edges e ∈ E such that the best response of ke is a−, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', ¯E = {e ∈ E : bke π = a−}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Consider a edge e ∈ ¯E and a vertex v ∈ V (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Since e ∈ ¯E the payment π(av) = 0 and no type ke plays action av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, if we modify the payments by letting π(av) = 1 2 on the action av we have three effects: i) the type ke changes the best response to av, ii) some other types e′ ∈ E could change from action a− to av,13 iii) the payment of type kv increases by 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Overall the principal’s total utility increases by 1 2λke since ke changes from action a− with payment 0 to action av with payment 1 2, and it decreases by − 1 2λkv as the payment to type kv increases by 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, if other types ke′, e′ ̸= e change from a− with payment 0 to action av with payment 1 2 the principal’s utility increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This implies that the principal’s utility does not decrease with this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, repeating this procedure we can build a payment function π with the same utility such that all the agent’s type plays actions av, v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, let ¯V be the set of vertexes with at least one agent of type ke, e ∈ E that plays this action, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', ¯V = {v ∈ V : bke π = av, e ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that ¯V is an vertex cover of size at most (1 + ǫ)ν, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that we can set payment π(av) = 0 for each v ∈ V \\ ¯V without decreasing the seller’s utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Removing the payment does not change any best response since the only type playing the action av is the type kv, the utility ukv(av) = 1, and the utility of playing any other action a ̸= av is ukv(a) + π(a) ≤ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, the principal’s utility is (ρ − | ¯V |) − 1 2| ¯V | + 1 2ℓ ρ + ℓ = ρ − 1 2| ¯V | + 1 2ℓ ρ + ℓ , since ρ − | ¯V | types kv, v ∈ V , play av and receive payment 0, | ¯V | types kv, v ∈ V , play av and receive payment 1 2, and all the types ke, e ∈ E, play an action av, v ∈ V and receive payment 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, since the principal’s utility is by assumption ρ− 1 2 | ¯V |+ 1 2 ℓ ρ+ℓ ≥ (1 − η) ρ− ν 2 + ℓ 2 ρ+ℓ , it holds | ¯V | ≤ η(2ρ + ℓ) + ν ≤ (1 + 7η)ν = (1 + 7η)ν = (1 + ε)ν, where the second inequality comes from ρ = 2 3ℓ and ℓ ≤ 3ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 13Notice that there could be other actions a′ v with π(av′) = 1 2 that provides the same utility of av for both the principal and the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 21 ARXIV PREPRINT - FEBRUARY 1, 2023 Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a posterior ξ ∈ ∆Θ, for any ρ ∈ (0, 1/2], there exists a payment function π : ∆Θ × A → R+ such that π(ξ, a) = β � θ∈Θ ξθ us θ(a) for every a ∈ A, where β ∈ [0, 1] is an (action-independent) parameter, and, additionally, the following holds: � k∈K λk � θ∈Θ ξθ � us θ(bk ξ,π) − π(s, bk ξ,π) � ≥ ρ � k∈K λk max a∈A � θ∈Θ ξθ � us θ(a) + uk θ(a) − uk θ(bk ξ) � − 2Ω(1/ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, such a parameter β is equal to 1 − 2−i for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' , ⌊1/2ρ⌋}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a first step, we notice that even if Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022a) show that linear contracts provide the desired approximation with respect to optimal contracts, their proof can be extended to show that linear contracts provide the same approximation with respect to the optimal social welfare, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', � k∈K λk maxa∈A[ra −ck a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' To prove this result, it is sufficient to follow all the steps of Theorem 3 of (Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2022a) except for the one in which Observation 1 is employed to upperboud the value of the optimal contract with the social welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, we can modify this result to hold in our setting exploiting Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' C Proofs Omitted from Section 5 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given any ǫ, α > 0, a posterior ξ∗ ∈ ∆Θ, and a payment function π : ∆Θ × A → R+, there always exists a signaling scheme γ ∈ ∆Ξq with q = 2 log(2m/α)/ǫ2 such that: � ξ∈Ξq γξ �� θ∈Θ ξθ us θ(bk,ǫ ξ,π) − π(ξ, bk,ǫ ξ,π) � ≥ � θ∈Θ ξ∗ θ us θ(bk ξ∗,π) − π(ξ∗, bk ξ∗,π) − α, for every buyer’s type k ∈ K, where we let π : ∆Θ ×A → R+ be a payment function that is optimal in every posterior ξ ∈ Ξq when the buyer plays an ǫ-best response, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', π solves Problem (4) for every ξ ∈ Ξq with bk ξ,π replaced by bk,ǫ ξ,π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Furthermore, the signaling scheme γ satisfies: � ξ∈Ξq γξ ξθ = ξ∗ θ ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let ˜ξ ∈ Ξq be the empirical mean of q i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' samples drawn according to ξ∗ ∈ ∆Θ, where each θ ∈ Θ has probability ξ∗ θ of being sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Therefore, ˜ξ ∈ Ξq is a random vector supported on q-uniform posteriors with expectation ξ∗ ∈ ∆Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, let γ ∈ ∆Ξq be a probability distribution such as, for each ξ ∈ Ξq, it holds γξ := Pr(˜ξ = ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We build a new payment function ˜π such that for each ξ ∈ ∆Θ and a ∈ A, we have ˜π(ξ, a) = π(ξ∗, a) Moreover, we let Ξq,ǫ be the set of posteriors such that ξ ∈ Ξq,ǫ if and only if for each a ∈ A it holds: ����� � θ∈Θ � ξθuk θ(a) − ξ∗ θuk θ(a) � ����� ≤ ǫ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (7) Then, for each ξ ∈ Ξq,ǫ, we have that Bk ξ∗,π ⊆ Bk,ǫ ξ,π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In particular, for any a∗ ∈ Bk ξ∗,π, ξ ∈ Ξq,ǫ and a ∈ A: � θ∈Θ ξθuk θ(a∗) + ˜π(ξ, a∗) ≥ � θ∈Θ ξ∗ θuk θ(a∗) + ˜π(ξ∗, a∗) − ǫ 2 (By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (7) and the definition of Bk ξ∗,π) ≥ � θ∈Θ ξ∗ θuk θ(a) + ˜π(ξ∗, a) − ǫ 2 ≥ � θ∈Θ ξθuk θ(a) + ˜π(ξ, a) − ǫ (By Equation (7)) which is precisely the definition of Bk,ǫ ξ∗,π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each a ∈ A, let ˜tk a := � θ∈Θ ˜ξθuk θ(a)+˜π(˜ξ, a) and tk a := � θ∈Θ ξ∗ θuk θ(a)+˜π(ξ∗, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By the Hoeffding’s inequality we have that, for each a ∈ A, Pr(|˜tk a − E[˜tk a]| ≥ ǫ 2) ≤ 2e−2q(ǫ/2)2 = 2e− log(2m/α) ≤ α m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (8) 22 ARXIV PREPRINT - FEBRUARY 1, 2023 Moreover, Equation (7) and the union bound yield the following: � ξ∈Ξq,ǫ γξ = Pr(˜ξ ∈ Ξq,ǫ) = Pr( � a∈A ��˜tk a − tk a �� ≤ ǫ 2) ≥ 1 − � a∈A Pr( ��˜tk a − tk a �� ≥ ǫ 2) ≥ 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (By Equation (8)) Let ¯ξ be a d-dimensional vector defined as ¯αθ := � ξ∈Ξq\\Ξq,ǫ γξξθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By definition and for the previous result we have: � θ∈Θ ¯ξθ ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' we can show: � ξ∈Ξq γξ � θ∈Θ ξθus θ(bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′) − π′(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′) ≥ � ξ∈Ξq γξ � θ∈Θ ξθus θ(bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − π(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) ≥ � ξ∈Ξq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ γξ � θ∈Θ ξθus θ(bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − π(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) ≥ � ξ∈Ξq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ γξ � θ∈Θ ξθus θ(bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − π(ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) (Bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π ⊆ Bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π for each ξ ∈ Ξq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ) = � θ∈Θ � us θ(bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − π(ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) �� � ξ∈Ξq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ γξξθ � = � θ∈Θ � us θ(bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − π(ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) �� � ξ∈Ξq γξξθ − ¯ξθ � (By definition of ¯α) = � θ∈Θ � ξθus θ(bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − π(ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) �� � ξ∈Ξq γξξθ � − � θ∈Θ � us θ(bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − π(ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) � ¯ξθ ≥ � θ∈Θ � us θ(bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − π(ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) �� � ξ∈Ξq γξξθ � − � θ∈Θ ¯ξθ (Utilities in [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 1]) ≥ � θ∈Θ ξ∗ θus θ(bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − π(ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, by definition of γ, we have that, for each θ ∈ Θ: � ξ∈Ξq γξξθ = ξ∗ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances where the buyer has limited liability and the number of buyer’s actions m is fixed, there exists a polynomial-time algorithm that, given a posterior ξ ∈ ∆Θ as input, computes the payments π(ξ, a) for a ∈ A of a payment function π : ∆Θ × A → R+ optimal in ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a posterior ξ ∈ ∆Θ and a tuple a ∈ A|K| we let Πa ⊆ Rm + be the set of payment functions π such that for each k ∈ K it holds ak ∈ Bk ξ,π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given an a ∈ A|K|, the problem of computing an optimal payment function restricted to payment functions in Πa can be formulated as follows: min π � k∈K λkπ(ξ, ak) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' � θ∈Θ ξθuk θ(ak) + π(ξ, ak) ≥ � θ∈Θ ξθuk θ(a′) + π(ξ, a′) ∀a′ ∈ A, k ∈ K π(ξ, a′) ≥ 0 ∀a′ ∈ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 23 ARXIV PREPRINT - FEBRUARY 1, 2023 We observe that, for each tuple a ∈ A|K|, the vertexes of the regions Πa ⊆ Rm + are identified by m of the common O(nm2 + m) constraints: � θ∈Θ ξθuk θ(a′) + π(ξ, a′) ≥ � θ∈Θ ξθuk θ(a′′) + π(ξ, a′′)∀a′ ̸= a′′ ∈ A, ∀k ∈ K π(, ξ, a′) ≥ 0 ∀a′ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the total number of vertexes defining all the regions Πa, a ∈ A|K|, is at most �nm2+m m � = O((nm2 + m)m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, since the objective function is linear in Πa for each tuple a ∈ A|K|, given the optimal tuple of induced actions a∗ ∈ A|K| the optimum is attained in one of the vertexes of Πa∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, there are overall �� � a∈A|K| V (Πa) �� = O((km2 + m)m) vertexes, where V (·) denotes the set of vertexes of the polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, when m is fixed, it is possible to enumerate in polynomial time over all the vertexes in � a∈A|K| V (Πa) and compute the optimal payment function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances where the buyer has limited liability and the number of buyer’s actions m is fixed, the problem of computing a seller-optimal protocol without menus admits a PTAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given two arbitrary constants α, ǫ > 0 we let (γ, π) be an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that an (α+2√ǫ)-optimal protocol (γ∗, π∗) can be computed in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a first step we define a signaling scheme γ∗ supported in Ξq as follows: γ∗ ˜ξ = � ξ∈supp(γ) γξγξ ˜ξ ∀˜ξ ∈ Ξq, where γξ ∈ ∆Ξq is the signaling scheme satisfying Lemma 6 with q = 2 log(2m/α) ǫ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First we observe that γ∗ ∈ ∆Ξq satisfies the consistency constraints, indeed we have: � ˜ξ∈Ξq γ∗ ˜ξ ˜ξθ = � ξ∈supp(γ) γξ � ˜ξ∈Ξq γξ ˜ξ ˜ξθ = � ξ∈supp(γ) γξξθ = µθ ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, let π∗ : ∆Θ × A → R+ be the optimal payment function in each ˜ξ ∈ Ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that the protocol (γ∗, π∗, 0) is (α + 2√ǫ)-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let π′′ : ∆Θ × A → R+ be the optimal payment function in each ξ ∈ Ξq when the buyer is playing an ǫ-best response, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', π′′(ξ, ·) ∈ arg max ˜π(ξ,·) � k∈K λk[ � θ ξθus θ(bk,ǫ ξ,˜π) − ˜π(ξ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, let π′ : ∆Θ ×A → R+ be the payment function such that π′(ξ, a) = (1−√ǫ)π′′(ξ, a)+√ǫ � θ∈θ ξθus θ(a) for each ξ and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' we have: � ˜ξ∈Ξq γ∗ ˜ξ � � θ∈Θ ˜ξθus θ(bk ˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π∗) − π∗(˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π∗) � ≥ � ˜ξ∈Ξq γ∗ ˜ξ � � θ∈Θ ˜ξθus θ(bk ˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′) − π′(˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′) � (Optimality of π∗) ≥ � ˜ξ∈Ξq γ∗ ˜ξ � � θ∈Θ ˜ξθus θ(bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′′) − π′′(˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′′) � − 2√ǫ (By Proposition 3) ≥ � ξ∈supp(γ) γξ � � ˜ξ∈Ξq γξ ˜ξ � � θ∈Θ ˜ξθus θ(bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′′) − π′′(˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′′) �� − 2√ǫ (By defintion of γ∗) ≥ � ξ∈supp(γ) γξ � � θ∈Θ ξθus θ(bk ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − π(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) � − α − 2√ǫ (By Lemma 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that the optimal payment π∗ : ∆Θ × A → R+ in each ξ ∈ Ξq can be computed in polynomial time employing Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, to compute the optimal signaling scheme γ∗ ∈ ∆Ξq we can solve the following LP: � k∈K λk � ξ∈Ξq γξ � θ∈Θ ξθus θ(bk ξ,π∗) − π∗(ξ, bk ξ,π∗) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 24 ARXIV PREPRINT - FEBRUARY 1, 2023 � ξ∈supp(γ) γξξθ = µθ ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Note that since |Ξq| = O(dq), all the payment function π∗ : Ξq × A → R+ can be precomputed in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the LP has polynomially many variables and constraints and can be solved efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, the solution returned by the LP is α + 2√ǫ-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances in which the buyer has limited liability, there exists an algorithm that, given any α, ǫ > 0 and ρ ∈ (0, 1/2] as input, returns a protocol without menus achieving a seller’s expected utility greater than or equal to ρ OPT − 2−Ω(1/ρ) − (α + 2√ǫ), where OPT is the seller’s expected utility in an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the algorithm runs in time polynomial in Ilog m—where I is the size of the problem instance and m is the number of buyer’s actions—, and the seller’s expected utility in the returned protocol is greater than or equal to OPTLIN − (α + 2√ǫ), where OPTLIN is the best expected utility achieved by a protocol parametrized by β as in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The proof follows the same steps of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' However, it relies on Theorem 4 instead of Lemma 7 to compute an approximate payment function for all q-uniform posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given any α > 0, a posterior ξ∗ ∈ ∆Θ, and a payment function π : ∆Θ × A → R+ that is optimal in every posterior ξ ∈ Ξq with q = ⌈9d/α2⌉, there exists a signaling scheme γ ∈ ∆Ξq: � ξ∈Ξq γξ �� θ∈Θ ξθ us θ(bk ξ,π) − π(ξ, bk ξ,π) � ≥ � θ∈Θ ξ∗ θus θ(bk ξ∗,π) − π(ξ∗, bk ξ∗,π) − α, for every receiver’s type k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Furthermore, the signaling scheme γ satisfies: � ξ∈Ξq γξ ξθ = ξ∗ θ ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We define a payment function π′ : ∆Θ ×A → R+ as follows: π′(ξ, a) = π(ξ∗, a) for each a ∈ A and ξ ∈ ∆Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Furthermore, we define: Iα(ξ∗) = � ξ ∈ ∆Θ : ∥ξ − ξ∗∥∞ ≤ α2 18d � , as the neighborhood of the given posterior ξ∗ ∈ ∆Θ and Ξ(ξ∗) = Iǫ(ξ∗) ∩ Ξq its intersection with the set Ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that if q ≥ 18d α2 , it holds ξ∗ ∈ co(Ξ(ξ∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 14 We show that for each ξ ∈ Iα(ξ∗), it holds bk ξ∗,π′ ∈ Bk,ǫ ξ,π′, where ǫ := α2/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a first step, by Hölder’s inequality we have that � θ∈Θ |(ξθ − ξ∗ θ)us θ(a)| ≤ d||ξ − ξ∗||∞ = ǫ/2 ∀a ∈ A, Moreover, by the definition of best response and the previous inequality, we have that: � θ∈Θ ξθuk θ(bk ξ∗,π′) + π′(ξ, bk ξ∗,π′) ≥ � θ∈Θ ξ∗ θuk θ(bk ξ∗,π′) + π′(ξ∗, bk ξ∗,π′) − ǫ/2 ≥ � θ∈Θ ξ∗ θuk θ(bk ξ,π′) + π′(ξ∗, bk ξ,π′) − ǫ/2 ≥ � θ∈Θ ξθuk θ(bk ξ,π′) + π′(ξ, bk ξ,π′) − ǫ ≥ � θ∈Θ ξθuk θ(a′) + π′(ξ, a′) − ǫ, for each a′ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This shows that bk ξ∗,π′ ∈ Bk,ǫ ξ,π′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let π∗ : ∆Θ × A → R+ be the payment function prescribed by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, we have that: 14Given a finite set A we denote with co(A) the set containing all the convex combination of elements in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 25 ARXIV PREPRINT - FEBRUARY 1, 2023 � θ∈Θ ξθus θ(bk ξ,π) − π(ξ, bk ξ,π) ≥ � θ∈Θ ξθus θ(bk ξ,π∗) − π∗(ξ, bk ξ,π∗) (Optimality of π′) ≥ � θ∈Θ ξθus θ(bk,ǫ ξ,π′) − π′(ξ, bk,ǫ ξ,π′) − 2√ǫ (By Proposition 3 ) ≥ � θ∈Θ ξθus θ(bk ξ∗,π′) − π′(ξ, bk ξ∗,π′) − 2√ǫ ≥ � θ∈Θ ξ∗ θus θ(bk ξ∗,π′) − π′(ξ∗, bk ξ∗,π′) − 2√ǫ − ǫ = � θ∈Θ ξ∗ θus θ(bk ξ∗,π′) − π′(ξ∗, bk ξ∗,π′) − 3√ǫ = � θ∈Θ ξ∗ θus θ(bk ξ∗,π) − π(ξ∗, bk ξ∗,π) − 3√ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This shows that the expected seller’s utility decreases of at most 3√ǫ when we consider sufficiently close posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, by Caratheodory’s theorem we can decompose ξ∗ as follows: � ξ′∈Ξ(ξ) γξ∗ ξ′ ξ′ θ = ξ∗ θ ∀θ ∈ Θ with γξ∗ ∈ ∆Ξ(ξ∗), where we recall that ξ∗ ∈ co(Ξ(ξ∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show now that such a decomposition decreases the expected seller’s utility only by the desired amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, we have that: � ξ′∈Ξ(ξ) γξ∗ ξ′ � � θ∈Θ ξ′ θus θ(bk ξ′,π) + π(ξ′, bk ξ′,π) � ≥ � ξ′∈Ξ(ξ) γξ∗ ξ′ � � θ∈Θ ξ∗ θus θ(bk ξ∗,π) + π(ξ, bk ξ∗,π) − 3√ǫ � = � θ∈Θ ξ∗ θus θ(bk ξ∗,π) + π(ξ∗, bk ξ∗,π) − 3√ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Since 3√ǫ ≤ α, this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances in which the buyer has limited liability and the number of states of nature d is fixed, there exists an algorithm that, given α > 0 and ρ ∈ (0, 1/2] as input, returns in polynomial time a protocol without menus achieving a seller’s expected utility greater than or equal to ρ OPT − 2−Ω(1/ρ) − α, where OPT is the seller’s expected utility in an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the seller’s expected utility in the returned protocol is greater than or equal to ρ OPTLIN − 2−Ω(1/ρ) − α where OPTLIN is the best expected utility achieved by a protocol parametrized by β as in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a constant α > 0 we let (γ, π) be an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a first step, we show that there exists a protocol (γ∗, π∗) achieving a seller’s expected utility of at least APX ≥ ρOPT−2−Ω(1/ρ)−α, where OPT is the utility achieved with (γ, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the payment function π∗ is a linear function with parameter β ∈ {1 − 2−i}i∈{i,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=',⌊ρ/2⌋}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We define a signaling scheme γ∗ supported in Ξq as follows: γ∗ ˜ξ = � ξ∈supp(γ) γξγξ ˜ξ ∀˜ξ ∈ Ξq, where γξ ∈ ∆Ξq is the signaling scheme satisfying Lemma 6 with q = 18d α2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First we observe that γ�� ∈ ∆Ξq satisfies the consistency constraints, indeed we have: � ˜ξ∈Ξq γ∗ ˜ξ ˜ξθ = � ξ∈supp(γ) γξ � ˜ξ∈Ξq γξ ˜ξ ˜ξθ = � ξ∈supp(γ) γξξθ = µθ ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, we can define as π∗ : ∆Θ × A → R+ as the payment function computed (in polynomial time) with Corollary 2 in each q-uniform posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let π′ be the optimal payment function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that the protocol (γ∗, π∗) achieves the desired approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: � ˜ξ∈Ξq γ∗ ˜ξ � � θ∈Θ ˜ξθus θ(bk ˜ξ,π∗) − π∗(˜ξ, bk ˜ξ,π∗) � 26 ARXIV PREPRINT - FEBRUARY 1, 2023 ≥ ρ \uf8eb \uf8ed � ˜ξ∈Ξq γ∗ ˜ξ � � θ∈Θ ˜ξθus θ(bk ˜ξ,π′) − π′(˜ξ, bk ˜ξ,π′) � \uf8f6 \uf8f8 − 2Ω(1/ρ) (Corollary 2) = � ξ∈supp(γ) γξ � � ˜ξ∈Ξq γξ ˜ξ � � θ∈Θ ˜ξθus θ(bk,ǫ ˜ξ,π) − π(˜ξ, bk,ǫ ˜ξ,π) �� (By defintion of γ∗) ≥ � ξ∈supp(γ) γξ � � θ∈Θ ξθus θ(bk ξ,π) − π(ξ, bk ξ,π) � − α (By Lemma 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This implies that since (γ∗, π∗) is feasible for the following LP, it has value at least ρOPT − 2Ω(1/ρ) − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' max γ≥0 � k∈K λk � ξ∈Ξq γξ � θ∈Θ ξθus θ(bk ξ,π∗) − π∗(ξ, bk ξ,π∗) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' � ξ∈supp(γ) γξξθ = µθ ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, to find the desired approximation it is sufficient to compute π∗ in each q-uniform posterior and solve the LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Note that since |Ξq| = O(qd), the computation of the payment function π∗ : ∆Θ × A → R+ and the computation of the previous LP require polynomial time for each fixed α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, to prove the second part of the statement it is sufficient to notice that π∗ is optimal with respect to the desired set of linear payment functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' D Proofs Omitted from Section 6 Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The problem of computing a seller-optimal protocol without menus is APX-hard, even when the number of buyer’s actions m is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We introduce a reduction from LINEQ-MA(1 − ζ, δ) to the design of the optimal protocol, showing that for ζ and δ small enough, the following holds: Completeness: If an instance of LINEQ-MA(1 − ζ, δ) admits a 1 − ζ fraction of satisfiable equations when variables are restricted to lie in the hypercube {0, 1}nvar, then there exists a protocol that provides to the seller’s expected utility at least of η, where η will be defined in the following;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Soundness: If at most a δ fraction of the equations can be satisfied, then every protocol provides to the seller’s expected utility at most η − c, where c is a constant defined in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In the rest of the proof, given a vector of variables x ∈ Qnvar, for i ∈ [nvar], we denote with xi the component corresponding to the i-th variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly, for j ∈ [neq], cj is the j-th component of the vector c, whereas, for i ∈ [nvar] and j ∈ [neq], the (j, i)-entry of A is denoted by Aji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Reduction As a preliminary step, we normalize the coefficients by letting ¯A := 1 τ A and ¯c := 1 τ 2 c, where we let τ := 2M max � maxi∈[nvar],j∈[neq] Aji, maxj∈[neq] cj, n2 var � and M will be defined in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It is easy to see that the normalization preserves the number of satisfiable equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, the number of satisfied equations of Ax = c is equal to the number of satisfied equations of ¯A¯x = ¯c, where ¯x = 1 τ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For every variable i ∈ [nvar], we define a state of nature θi ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, we introduce three additional states θ0, θ1, θ2 ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The prior distribution µ ∈ int(∆Θ) is defined in such a way that µθi = 1 2n2var for every i ∈ [nvar], while µθ0 = nvar−1 2nvar , µθ1 = 1 4, and µθ2 = 1 4 (notice that � θ∈Θ µθ = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We define four buyer’s types k1 j , k2 j , k3 j , k4 j ∈ K for each equation j ∈ [neq], where the probability of observing each buyer’s type is 1 8neq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, we define an additional type k⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' All the types k ∈ K have budget bk = ν/2, where ν will be defined in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The buyer has 9 actions available, namely A := {a0, a1, a2, a3, a4, a5, a6, a7, a8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, we define the utilities of the players, where the utility is 0 when not specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each k1 j , j ∈ [neq], the utilities are: u k1 j θi (a0) = 1 2 for each i ∈ [nvar], 27 ARXIV PREPRINT - FEBRUARY 1, 2023 u k1 j θi (a1) = 1 2 − ¯Aji + ¯cj for each i ∈ [nvar], u k1 j θi (a2) = 1 2 + ¯Aji − ¯cj for each i ∈ [nvar] u k1 j θ0 (a0) = 1 2, u k1 j θ0 (a1) = 1 2 + ¯cj, u k1 j θ0 (a2) = 1 2 − ¯cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' u k1 j θ1 (a3) = 1 2 + 2ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each k2 j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' j ∈ [neq],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' the utilities are: u k2 j θi (a0) = 1 2 − ¯Aji + ¯cj for each i ∈ [nvar],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' u k2 j θi (a7) = 1 2 for each i ∈ [nvar] u k2 j θ0 (a0) = 1 2 + ¯cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' u k2 j θ1 (a3) = 1 2 + 2ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each type k3 j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' j ∈ [neq] the utilities are: u k3 j θi (a0) = 1 2 + ¯Aji − ¯cj for each i ∈ [nvar],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' u k3 j θi (a7) = 1 2 for each i ∈ [nvar] u k3 j θ0 (a1) = 1 2 − ¯cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' u k3 j θ1 (a3) = 1 2 + 2ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each type k4 j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' j ∈ [neq] the utilities are equivalent to the one of type k1 j but with the following differences: ukj θ (a5) = 1 2 for each θ ∈ Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ukj θi (a7) = 1 2 for each i ∈ [nvar],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' the utilities of type k⋆ are: uk⋆ θ1 (a6) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' uk⋆ θ (a1) = 1 for each θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, we let uk θ(a8) = 1 2 for every k ∈ K and θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, the utility of the seller is: us θ(a6) = 1 4 for each θ ∈ Θ, us θ(a5) = 4ν for each θ ∈ Θ, 28 ARXIV PREPRINT - FEBRUARY 1, 2023 us θ(a0) = ν for each θ ∈ Θ, us θ(a7) = 2ν for each θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We recall that the utility is 0 when not defined explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Suppose that there exists a vector ˆx ∈ {0, 1}nvar such that at least a fraction 1 − ζ of the equations in Aˆx = c are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let X1 ⊆ [nvar] be the set of variables i ∈ [nvar] with ˆxi = 1, while X0 := [nvar] \\ X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given the definition of ¯A and ¯c, there exists a vector ¯x ∈ {0, 1 τ }nvar such that at least a fraction 1 − ζ of the equations in ¯A¯x = ¯c are satisfied, and, additionally, ¯xi = 1 τ for all the variables in i ∈ X1, while ¯xi = 0 whenever i ∈ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let us consider an (indirect) signaling scheme φ : Θ → ∆S where the set of signals is S := {s1, s2, s3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let q := nvar(nvar−1) τ−|X1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each i ∈ [nvar], let φθi(s1) = q and φθi(s2) = 1 − q if i ∈ X1, while φθi(s2) = 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, let φθ0(s1) = 1, φθ1(s3) = 1 and φθ2(s2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, all the other probabilities φθ(s) are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It is easy to see that the signaling scheme is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, we set the price p = ν/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, we set π(s3, a6) = 2ν and all the other payments π(s, a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Now, we compute the expected seller’s utility due of each type of buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The buyer of type k⋆ in the posterior ξs3 plays the action a6 and gets utility � θ∈θ ξs3 θ uk⋆ θ (a6) + π(s3, a6) = 1 + 2ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, in the other posteriors ξs1 and ξs2 the seller’s utility is at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, the protocol is IR for the buyer since the expected utility declining the protocol is 1 while accepting it is −π/2 + 1 · 3 4 + (1 + 2ν) 1 4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the expected principal utility when the buyer’s type is k⋆ is at least � s∈S � θ∈θ ξs θuk⋆ θ (bk⋆ ξs,π) = 1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Consider a buyer k1 j , j ∈ [neq], such that the j-th equality is satisfied by the vector ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Now, let us take the buyer’s posterior ξs1 ∈ ∆Θ induced by the signal s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let h := q n2var � i∈X1 q n2var + nvar−1 nvar .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, using the definition of ξs1, it is easy to check that ξ1 θi = h for every i ∈ Xs1, ξs1 θi = 0 for every i ∈ X0, while ξs1 θ0 = nvar−1 nvar � i∈X1 q n2var + nvar−1 nvar = 1 − h ��X1��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The buyer of type kj ∈ K experiences a utility of � θ∈Θ ξs1 θ ukj θ (a0) = 1 2 by playing action a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Instead,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' the utility she gets by playing a1 is defined as follows: � θ∈Θ ξs1 θ ukj θ (a1) = � i∈X1 h �1 2 − ¯Aji + ¯cj � + ξ1 θ0 �1 2 + ¯cj � = = h ��X1�� �1 2 + ¯cj � − h � i∈X1 ¯Aji + � 1 − h ��X1��� �1 2 + ¯cj � = = 1 2 + ¯cj − h � i∈X1 ¯Aji = 1 2 + ¯cj − 1 τ � i∈X1 ¯Aji = 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' where the second to last equality holds since h = 1 τ (by definition of h and q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' while the last equality follows from the fact that the j-th equation is satisfied,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 1 τ � i∈X1 ¯Aji = ¯cj (recall that ¯xi = 1 τ for all i ∈ X1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Using similar arguments, we can write � θ∈Θ ξs1 θ ukj θ (a2) = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, all the other actions have utility 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the buyer plays a0 in the posterior ξs1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In posterior ξs2 induced by signal s2, the utility of each action different from a8 is strictly smaller than 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the buyer will play a8, while in posterior ξs3 induced by signal s3, the utility of action a3 is 1 2 + 2ν and the buyer will play a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence the expected utility of the buyer is 3 4 1 2 + 1 4( 1 2 + 2ν) − p = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the protocol is IR for the buyer since if she declines the protocol the utility is 1 2 while if she accepts the protocol the utility is 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, when the buyer’s type is k1 j the expected seller’s utility is νµθ0 + ν/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Consider a buyer k2 j or k3 j , j ∈ neq such that the j-th equality is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A similar argument as before shows that in posterior ξs1 the buyer’s optimal action is a7, while in posterior ξs2, the optimal action is a8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In posterior ξs3, the optimal action is a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the expected buyer’s utility is 3 4 1 2 + 1 4( 1 2 + 2ν) − p = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the protocol is IR for the buyer and provides expected seller’s utility at least 2νµθ0 + ν/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Consider a buyer k4 j , j ∈ [neq] such that the j-th equality is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The buyer has an utility similar to k1 j and plays the same best responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, it is indifferent in participating or not participating to the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 29 ARXIV PREPRINT - FEBRUARY 1, 2023 We assume that they brake ties in favor of the seller and does not accept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' She plays action a5 and the expected seller’s utility is 2ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Since all the other buyer’s types provide positive utility —it never happens that the expected payment from the seller to the buyer exceeds the payment from the buyer to the seller—, the expected seller’s utility is at least η = 1 32 + (1 − ζ)1 8(µθ0ν + ν/2) + (1 − ζ)1 4(µθ02ν + ν/2) + (1 − ζ)1 82ν Soundness As a first step, we upperbound the expected seller’s utility from each type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It is easy to see that the maximum expected utility that the seller can extract from the buyer’s type k⋆ is at most 1 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the maximum expected utility that the seller can extract from a buyer of type k1 j , j ∈ [neq], is at most 1 8(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The maximum expected utility that the seller can extract from a buyer of type k2 j or k3 j , j ∈ [neq] is at most 1 4 3 2ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, the maximum expected utility that the seller can extract from a buyer of type k4 j , j ∈ [neq], is 1 82ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Using the previous upperbounds, we can bound the component of the utility due to each set of types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each constant t < 1, there exist constants c = c(t), ζ = ζ(t) such that if the expected utility is greater than η − c then the expected utility from types k1 j , j ∈ [neq], is at least t 1 8(ν), the expected utility from types k2 j , j ∈ [neq], and k3 j , j ∈ [neq], is at least t 1 4 3 2ν, and the expected utility from types k4 j , j ∈ [neq], is at least t 1 82ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' To see that, consider for instance the types k1 j , j ∈ [neq].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It must hold: 1 32 + ¯t1 8ν + 1 4 3 2ν + 1 82ν ≥ 1 32 + (1 − ζ)1 8(µθ0ν + ν/2) + (1 − ζ)1 4(µθ02ν + ν/2) + (1 − ζ)1 82ν Since for nvar large enough µθ0 is close to 1 2, for c(t), ζ(t) small enough constant the equation is satisfied for ¯t ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A similar result holds for every other set of types k2 j with j ∈ [neq], k3 j with j ∈ [neq], and k4 j with j ∈ [neq].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The next step is to show the existence of a posterior in which a t fraction of agent of types k1 j , j ∈ [neq], play a0 and the the same holds for each other set of types k2 j ,k3 j with action a7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Suppose by contradiction that there is no posterior in which a t fraction of k1 j , j ∈ [neq], plays a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First, notice that the maximum payment is at most p = ν/2+ 1 M , otherwise all the buyer’s types k1 j are not IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the seller’s utility minus payment is greater than 0 in a posterior only if the agent plays a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, it is easy to see that it is sufficient to consider signaling schemes that induce posteriors such that if ξθi > 0, then ξθ1 = 0 and ξθ2 = 0 since states ξθ1 and ξθ2 disincentivize the actions with high seller’s utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the maximal utility from agents of types k1 j is at most 1 8 � ν/2 + 1 M + (t − 1/neq)1 2ν � < t1 8ν, for M large enough, reaching a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A similar argument holds for the other types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This implies that there exists a set Q ⊆ [neq] and a posterior ξ such that for each j ∈ Q all the buyers k1 j , k2 j , and k3 j in the posterior play a0,a7, and a7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that |Q| ≥ 1 − 3(1 − t) and for t large enough |Q| > δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Suppose that there exists a signal inducing a posterior ξ ∈ ∆Θ in which all the buyer’s types k1 j , j ∈ Q best respond by playing action a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that there exists at least one j ∈ Q such that it holds � θ∈Θ ξθu k1 j θ (a1) > � θ∈Θ ξθu k1 j θ (a0) or � θ∈Θ ξθu k1 j θ (a2) > � θ∈Θ ξθu k1 j θ (a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For every buyer’s type k1 j ∈ K, it holds � θ∈Θ ξθukj θ (a0) = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, it is the case that: � θ∈Θ ξθu k1 j θ (a1) = � i∈[nvar] ξθi �1 2 − ¯Aji + ¯cj � + ξθ0 �1 2 + ¯cj � = 1 2 + ¯cj − � i∈[nvar] ξθi ¯Aji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly, it holds: � θ∈Θ ξθu k1 j θ (a2) = 1 2 − ¯cj + � i∈[nvar] ξθi ¯Aji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Suppose by contradiction that for every type k1 j , j ∈ Q, it is the case that � θ∈Θ ξθu k1 j θ (a0) ≥ � θ∈Θ ξθu k1 j θ (a1), which implies that ¯cj − � i∈[nvar] ξθi ¯Aji ≤ 0, whereas it holds � θ∈Θ ξθu k1 j θ (a0) ≥ � θ∈Θ ξθukj θ (a2), implying −¯cj + � i∈[nvar] ξθi ¯Aji ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, � i∈[nvar] ξθi ¯Aji = ¯cj for every j ∈ Q and the vector ˆx ∈ Qnvar with ˆxi = ξθi for all i ∈ [nvar] satisfies at least a fraction δ of the equations, reaching a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Since we 30 ARXIV PREPRINT - FEBRUARY 1, 2023 have that t types k1 j play a0, this implies that π(ξ, a0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' However, at the same time we have that the buy- ers of type k2 j and k3 j plays action a7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Consider a j∗ ∈ Q such that � θ∈Θ ξθu k1 j∗ θ (a1) > � θ∈Θ ξθu k1 j∗ θ (a0) or � θ∈Θ ξθu k1 j∗ θ (a2) > � θ∈Θ ξθu k1 j∗ θ (a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Recall that this buyer must play a7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' If the first inequality holds then it must hold � θ∈Θ ξθu k2 j∗ θ (a7) + π(ξ, a7) ≥ � θ∈Θ ξθu k2 j∗ θ (a0) + π(ξ, a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, � θ∈Θ ξθu k2 j∗ θ (a7) = � θ∈Θ ξθu k1 j∗ θ (a0) < � θ∈Θ ξθu k1 j∗ θ (a1) = � θ∈Θ ξθu k2 j∗ θ (a0), implying π(ξ, a7) > π(ξ, a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A similar argument holds for the buyer k3 j∗ if the second inequality is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This implies that type k4 j∗ can play the same best responses of player k1 j in any posterior different from ξ and play action a7 in ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the expected utility of buyer k4 j∗ is strictly greater than the one of k1 j∗ (that is IR), and hence it is strictly IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We conclude the proof showing that the utility of this buyer’s type is too small, reaching a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First, notice that the seller must induce a posterior with ξθ1 ≥ 3 4 with probability at least 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In all the other posteriors the seller’s utility from type k∗ is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' However, it must hold that the utility from type k⋆ is at least 1 64 for ν small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, playing posteriors with ξθ1 ≥ 3 4 with probability smaller than 1 8 the seller’s utility form type k∗ is at most 1 2 1 4 1 8 < 1 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Now consider the type k4 j∗ that is IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In a posterior ξ with ξθ1 ≥ 3 4, the seller’s utility when the type is k4 j∗ is at most 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the total utility from this type is at most p + 7 84ν ≤ ν + 1/M, where the last inequality follows by the fact that the payment is at most ν 2 + 1/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For |Q| large enough, we have that a |Q|/neq − δ fraction of types k4 j provide seller’s utility at most ν + 1/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the total utility from type k4 j is at most 1 8[(|Q|/neq − δ)(ν + 1/M) + (1 − (|Q|/neq − δ))2ν] ≤ t 1 82ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, we reach a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' There exists an algorithm that, given any α > 0 and ρ ∈ (0, 1/6] as input, computes a protocol without menus whose seller’s expected utility is greater than or equal to ρ OPT − 2−Ω(1/ρ) − α, where OPT is the seller’s expected utility in an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the algorithm runs in time polynomial in Ilog m—where I is the size of the problem instance—when it is implemented with the algorithm in Theorem 7 as a subroutine, while it runs in time polynomial in Id when it is implemented with the algorithm in Theorem 8 as a subroutine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let (φ, p, π) be an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, the seller’s expected utility is given by: � k /∈Rφ,p,π λk � θ∈Θ µθus θ(bk µ) + � k∈Rφ,p,π λk �� s∈S � θ∈Θ µθφθ(s) � us θ(bk ξs,π) − π(s, bk ξs,π) � + p � , where we recall that Rφ,p,π is the set of buyer’s types for which the IR constraint is satisfied under protocol (φ, p, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a signal s ∈ S and a type k ∈ K, let bk ξs ∈ arg maxa∈A � θ∈Θ µθφθ(s)uk θ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, bk ξs is an opti- mal action for the buyer without considering the payment function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, the seller’s utility can be spitted in three components: (i) The utility from the buyer’s types that are not IR U1 := � k /∈Rφ,p,π λk � θ∈Θ µθus θ(bk µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (ii) The maximum seller’s utility deriving from the buyer’s action U2 := � k∈Rφ,p,π λk �� s∈S � θ∈Θ µθφθ(s) � us θ(bk ξs) + uk θ(bk ξs,π) − uk θ(bk ξs) � � , where we use the fact that to incentivize action bk ξs,π over bk ξs the payment must be at least � s∈S � θ∈Θ µθφθ(s)(uk θ(bk ξs)−uk θ(bk ξs,π)) � θ∈Θ µθφθ(s) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (iii) The utility related to the overall payment that the seller’s can extract from the buyer given the price function π U3 := � k∈Rφ,p,π λk � p − � s � θ µθφθ(s) � π(s, bk s,π) + uk θ(bk s,π) − uk θ(bk ξs) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 31 ARXIV PREPRINT - FEBRUARY 1, 2023 Notice that the term U2 + U3 is the utility deriving from buyer’s types for which the IR constraint is satisfied, where we add, respectively subtract, the term � k∈Rφ,p,π λk �� s∈S � θ∈Θ µθφθ(s) � uk θ(bk ξs,π) − uk θ(bk ξs) � � to U2, respectively U3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In the following, we design three protocols (φ1, p1, π1), (φ2, p2, π2), and (φ3, p3, π3), each with seller’s utility that approximates the corresponding utility terms U1, U2, and U3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We will show that this will implies that at least one protocol provides a good approximation of the overall seller’s utility, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', of U1 + U2 + U3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Approximate U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The protocol (φ1, p1, π1) that provides no information, charges no price, and does not provides any payment has seller’s utility � k∈K � θ µθus θ(bk µ) ≥ � k /∈Rφ,p,π � θ µθus θ(bk µ) = U1 Approximate U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By Corollary 2, we know that for each signal s ∈ S (inducing a posterior ξs) and ρ ∈ (0, 1/2], there exists a linear contract π′(s, ·) such that π′(s, a) = β � θ∈Θ ξs θus θ(a) with parameter β = 1 − 2−i, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' , ⌊ 1 2ρ⌋} that guarantees: � k∈K λk � � θ∈Θ ξs θ � us θ(bk ξs,π′) − π′(ξs, bk ξs,π′) � � (11a) ≥ ρ � k∈K λk �� θ ξs θ � us θ(bk ξs,π) + uk θ(bk ξs,π) − uk θ(bk ξs) � � − 2−Ω(1/ρ) (11b) ≥ ρ � k∈Rφ,p,π λk �� θ ξs θ � us θ(bk ξs,π) + uk θ(bk ξs,π) − uk θ(bk ξs) � � − 2−Ω(1/ρ) (11c) where the first inequality comes from Corollary 2, and the last one since we restrict the elements in the first summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Now, we need a protocol (φ2, p2, π2) that approximate the utility obtained by the optimal protocol that uses only linear payment functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' When the number of states is fixed, we can approximate the optimal protocol that uses linear payment functions using Theorem 8 with an additive loss α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Otherwise, we can use Theorem 7 that is polynomial time when the number of actions is fixed, while it runs in quasi-polynomial time and provides a loss α when instantiated with sufficiently small parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, protocol (φ2, p2, π2) can be computed in time poly(min{Id, Ilog(m)}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that both the algorithms returns a protocol such that p = 0 and hence p2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' we can show that the protocol (φ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' π2) has seller’s utility � k∈K λk �� s∈S � θ µθφθ(s) � us θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π2) − π2(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π2) � � ≥ � k∈K λk �� s∈S � θ µθφθ(s) � us θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′) − π′(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′ � � − α = � k∈K λk �� s∈S �� θ µθφθ(s) � � θ ξs θ � us θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′) − π′(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′ � � − α = � s∈S �� θ µθφθ(s) � � k∈K λk �� θ ξs θ � us θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′) − π′(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′ � � − α ≥ � s∈S �� θ µθφθ(s) � � ρ � k∈R λk � θ ξs θ � us θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) + uk θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − uk θ(bk ξs) � − 2−Ω(1/ρ) � − α = � s∈S �� θ µθφθ(s) � \uf8ee \uf8f0ρ � k∈Rφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π λk � θ ξs θ � us θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) + uk θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − uk θ(bk ξs) � \uf8f9 \uf8fb − 2−Ω(1/ρ) − α 32 ARXIV PREPRINT - FEBRUARY 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2023 = ρ � k∈Rφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π λk � s∈S �� θ µθφθ(s) � �� θ ξs θ � us θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) + uk θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − uk θ(bk ξs) � � − 2−Ω(1/ρ) − α = ρ � k∈Rφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π λk � s∈S � θ µθφθ(s) � us θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) + uk θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − uk θ(bk ξs) � − 2−Ω(1/ρ) − α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' where the first inequality holds since π′ employs linear payments functions and (φ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' π2) has an additive loss α w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' any protocol that employs linear payments functions, while the second inequality comes from Equation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Approximate U3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let δk := � θ µθuk θ(bk θ) − � θ µθuk θ(bk µ) for each k ∈ K, where bk θ is the best response of agent of type k ∈ K when the state of nature is θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each k ∈ Rφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' by the definition of IR it holds � s∈S � θ∈Θ µθφθ(s)[π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) + uk θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π)] − p ≥ � θ µθuk θ(bk µ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (12) Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' p − � s∈S � θ∈Θ µθφθ(s) � π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) + uk θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − uk θ(bk ξs) � = p − � s∈S � θ∈Θ µθφθ(s) � π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) + uk θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) � + � s∈S � θ∈Θ µθφθ(s)uk θ(bk ξs) ≤ p − � s∈S � θ∈Θ µθφθ(s) � π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) + uk θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) � + � s∈S � θ∈Θ µθφθ(s)uk θ(bk θ) = p − � s∈S � θ∈Θ µθφθ(s) � π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) + uk θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) � + � θ∈Θ µθuk θ(bk θ) ≤ − � θ∈Θ µθuk θ(bk θ) + � θ∈Θ µθuk θ(bk θ) ≤ δk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' where the first inequality follows by the optimality of action bk θ in state θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' and the second one by Equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Next, we show that for each ζ ∈ [0, 1] we can design a protocol with seller’s utility of at least ζ 2 � k∈K δk − 2−1/ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let Pζ := {2−i}i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=',⌊1/ζ⌋} ∪ {0}, and for each k ∈ K let pk be the greatest p ∈ Pζ such that p ≤ δk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, � k∈K λkpk ≥ � k∈K λk � δk/2 − 2−⌊1/ζ⌋� = � k∈K λkδk/2 − 2−⌊1/ζ⌋, where the inequality holds since either pk ≥ δk/2 or pk ≤ 2−⌊1/ζ⌋ Hence, � p∈Pζ p � k∈K:pk=p λk ≥ � k∈K λkδk/2 − 2−⌊1/ζ⌋, implying max p∈Pζ p � k∈K:pk=p λk ≥ 1 2|Pζ| � k∈K λkδk − 2−⌊1/ζ⌋ ≥ ζ 2 � k∈K λkδk − 2−⌊1/ζ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let p∗ = argmaxp∈Pζ p � k∈K:pk=p λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Consider the protocol (φ3, p3, π3) that charges payment p3 = p∗, reveals all information with φ3 and set payment π3(s, a) = 0 for each s ∈ S and a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that this protocol satisfies the IR constraint for all the players such that pk = p∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, for all these types it holds � θ∈Θ µθuk θ(bk θ) − p∗ ≥ � θ∈Θ µθuk θ(bk θ) − δk = � θ∈Θ µθuk θ(bk θ) − �� θ µθuk θ(bk θ) − � θ µθuk θ(bk µ) � ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (14) Then, the utility of the protocol is at least the payment obtained by the buyers’ type in Rφ3,p3,π3 ⊇ {k ∈ K : pk = p∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In particular, it is at least p∗ � k∈K:pk=p∗ λk ≥ ζ 2 � k∈K λkδk − 2−⌊1/ζ⌋ 33 ARXIV PREPRINT - FEBRUARY 1, 2023 ≥ ζ 2 � k∈Rφ,p,π λkδk − 2−⌊1/ζ⌋ ≥ ζ 2 � k∈Rφ,p,π λk � p − � s∈S � θ µθφθ(s)[π(s, bk ξs,π) + uk θ(bk ξs,π) − uk θ(bk ξs)] � − 2−⌊1/ζ⌋ = ζ 2U3 − 2−⌊1/ζ⌋, where in the the first inequality we use Equation (14), and in the third inequality we use Equation (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Equivalently, setting ρ = ζ/2, we obtain that for each ρ ∈ [0, 1/2] there exists a protocol (φ3, p3, π3) that has seller’s utility at least ρU3 − 2−Ω(1/ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Wrapping up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let i = arg maxj∈{1,2,3} Uj and OPT be the seller’s utility with the optimal protocol (φ, p, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, since U1+U2+U3 = OPT, we have that Ui ≥ 1 3OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, since for each ρ ∈ [0, 1/2] we can approximate each utility Ui, i ∈ {1, 2, 3} with a protocol with utility at least ρUi −2−Ω(1/ρ) −α, the seller’s utility of our approximation algorithm is at least ρUi − 2−Ω(1/ρ) − α ≥ ρOPT/3 − 2−Ω(1/ρ) − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, setting ρ′ = ρ/3, we obtain that for each ρ′ ∈ [0, 1/6] the utility of the designed protocol is at least OPT − 2−Ω(1/ρ) − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a seller’s protocol without menus, there always exists another protocol without menus which is generalized-direct and generalized-persuasive, and achieves the same seller’s expected utility as the original protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let (φ, π, p) be a protocol and let be s1, s2 ∈ S be two signals such that bk ξs1 = bk ξs2 for each receiver’s type k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that it is always possible to define a new protocol (φ∗, π∗, p) that employs a single signal s∗ instead of s1 and s2 achieving the same seller’s expected utility while satisfying the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, we define a new signaling scheme φ∗ as follows: �φ∗ θ(s∗) = φθ(s1) + φθ(s2) ∀θ ∈ Θ φ∗ θ(s) = φθ(s) ∀θ ∈ Θ, ∀s ∈ S \\ {s1, s2} and a new payment function π∗ as follows: �π∗(s∗, a) = zπ(s1, a) + (1 − z)π(s2, a) ∀a ∈ A π∗(s, a) = π(s, a) ∀a ∈ A, ∀s ∈ S \\ {s1, s2} with z = � θ∈Θ µθφθ(s1)/(� θ∈Θ µθ(φθ(s1) + φθ(s2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a first step, we observe that for each k ∈ K it holds: � θ∈Θ µθ � φθ(s1) � us θ(bk ξs1,π) − π(s1, bk ξs1,π) � + φθ(s2) � us θ(bk ξs2,π) − π(s2, bk ξs2,π) � � = � θ∈Θ µθφ∗ θ(s∗) � us θ(bk ξs∗,π∗) − π∗(s∗, bk ξs∗,π∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, for each k ∈ K it holds: � θ∈Θ µθ � φθ(s1) � uk θ(bk ξs1 ,π) + π(s1, bk ξs1,π) � + φθ(s2) � uk θ(bk ξs2,π) + π(s2, bk ξs2,π) � � = � θ∈Θ µθφ∗ θ(s∗) � uk θ(bk ξs∗,π∗) + π∗(s∗, bk ξs∗,π∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, noticing that for each signal s ∈ S \\ {s1, s2} the seller’s utility and the buyer’s utility does not change from (φ, π, p) to (φ∗, π∗, p), the set R of buyer’s type for which the IR is satisfied does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a consequence, the two protocols achieve the same seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Applying this procedure to all the couples of signals that induces the same vector of best responses, we obtain a generalized-direct and generalized-persuasive protocol providing the same seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a protocol without menus, there always exists another protocol (φ, p, π) such that p = bk for some k ∈ K, while achieving the same seller’s expected utility as the original protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 34 ARXIV PREPRINT - FEBRUARY 1, 2023 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let (φ, π, p) be a protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that there exists a ˆk ∈ K and a payment function ˆπ such that the protocol (φ, ˆπ, bˆk) provides the same seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let ˆk ∈ arg min k∈Rφ,π,p:bk≥p{bk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We observe that all the buyer’s types k ∈ Rφ,π,p have enough budget to participate in the protocol,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', bk ≥ bˆk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Furthermore, we define ˆπ(s, a) = π(s, a) + bˆk − p for each s ∈ S and a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, we show that the set of types Rφ,π,p = Rφ,ˆπ,ˆp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, for each type k ∈ Rφ,ˆπ,ˆp it holds � θ∈Θ � s∈S µθφθ(s) � uk θ(bk ξs,ˆπ) + ˆπ(s, bk ξs,ˆπ) � − bˆk = � θ∈Θ � s∈S µθφθ(s) � uk θ(bk ξs,ˆπ) + π(s, bk ξs,ˆπ) + bˆk − p � − bˆk = � θ∈Θ � s∈S µθφθ(s) � uk θ(bk ξs,π) + π(s, bk ξs,π) � − p, and hence k ∈ Rφ,π,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly, we can prove that each buyer’s type k /∈ Rφ,ˆπ,ˆp does not belong to Rφ,π,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It follows that Rφ,π,p = Rφ,ˆπ,ˆp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, we can show that the seller’s utility results equal to the one in (φ, π, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, we have: � k∈Rφ,p,π λk � � θ∈Θ � s∈S µθφθ(s) � us θ(bk ξs,π) − π(s, bk ξs,π) � + p � + � k /∈Rφ,p,π λk � θ∈Θ µθus θ(bk µ) = � k∈Rφ,ˆ p,ˆπ λk � � θ∈Θ � s∈S µθφθ(s) � us θ(bk ξs,ˆπ) − ˆπ(s, bk ξs,ˆπ) � + bˆk � + � k /∈Rφ,ˆ p,ˆπ λk � � θ∈Θ µθus θ(bk µ) � This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances in which the number of buyer’s types n is fixed, the problem of computing a seller-optimal protocol without menus admits a polynomial-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In the following, we present an algorithm to compute an optimal protocol that works in polynomial time when the number of buyer’s types is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a first step, we observe that, thanks to Lemma 10, the initial payment required by the seller coincides with bk for some k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Furthermore, we can focus on direct protocols by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, given a price p ∈ {bk}k∈K and a set of buyer’s types R ⊆ K ∩ {k ∈ K : bk ≥ p} for which the IR constraint is satisfied, the the problem of computing the optimal protocol can be formulated as Problem (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly to Section 3, we can provide a linear relaxation of Problem (6) introducing a variable l(a, a′) that replaces � θ∈Θ µθφθ(a)π(a, a′) for each a ∈ An and a′ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, we obtain the following LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' max φ≥0,l≥0 � k∈R λk � a∈An �� θ∈Θ µθφθ(a)us θ(ak) − l(a, ak) � + � k /∈R λk � θ∈Θ µθus θ(bk µ) (15a) � θ∈Θ µθφθ(a)uk θ(ak) + l(a, ak) ≥ � θ∈Θ µθφθ(a)uk θ(a′) + l(a, a′) ∀k ∈ R, ∀a ∈ An, ∀a′ ̸= ak ∈ A (15b) � a∈An �� θ∈Θ µθφθ(a)uk θ(ak) + l(a, ak) � − bk ≥ � θ∈Θ µθuk θ(bk µ) ∀k ∈ R (15c) � a∈An �� θ∈Θ µθφθ(a)uk θ(ak) + l(a, ak) � − bk ≤ � θ∈Θ µθuk θ(bk µ) ∀k ̸∈ R (15d) � a∈An φθ(a) = 1 ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (15e) Hence, once we fix bk and R, LP (15) returns a solution that has the same value of the optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 35 ARXIV PREPRINT - FEBRUARY 1, 2023 To compute the optimal protocol we can iterate over all the possible prices p ∈ {bk}k∈K and all the possible subsets R ⊆ K ∩ {k ∈ K : bk ≥ p} of receivers types for which the IR constraint is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that, given a price p, the IR constraint can be satisfied only the buyer’s type k ∈ K with bk ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, we solve LP (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, we return the solution with highest value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As we show in the first part of the proof, this solution has the same value of the optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, it is easy to check that the overall procedure requires to solve O(n2n) LPs, showing that the algorithm runs in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' To conclude the proof, we need to show how to modify the solution of LP 15 to obtain a protocol, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', a solution to Problem (6), with at least the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' To do so, we exploit a similar approach to the one presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let (φ, l) be the solution to LP (15) returned by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Suppose that there exists a couple (¯a, ¯k) such that l(¯a, a¯k) > 0 and � θ∈Θ µθφθ(¯a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show how to obtain a solution such that l(¯a, a) = 0 for each a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that by Constraint (15b), it holds l(¯a, ¯ak) ≥ l(¯a, a) for each k ∈ K, a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This implies that l(¯a, ¯ak) = l(¯a, ¯ak′) for each k ̸= k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We denote this value with l(¯a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let ˆa ∈ An be any signal such that � θ∈Θ µθφθ(ˆa) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Consider a assignment (φ, l′) to the variables such that l′(¯a, a) = 0 for each a ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' l′(ˆa, a) = l(ˆa, a) + l(¯a) for each a ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' l′(a) = l(a) for each a /∈ {¯a, ˆa}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that this solution is feasible to LP (15) and has the same objective value of (φ, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' it holds � k∈R λk � a∈An �� θ∈Θ µθφθ(a)us θ(ak) − l′(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ak) � + � k /∈R λk � θ∈Θ µθus θ(bk µ) = � k∈R λk � � a∈An\\{¯a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ˆa} �� θ∈Θ µθφθ(a)us θ(ak) − l′(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ak) � + � θ∈Θ µθφθ(¯a)us θ(¯ak) + � θ∈Θ µθφθ(ˆa)us θ(ˆak) − (l(ˆa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ˆak) − l(¯a)) � + � k /∈R λk � θ∈Θ µθus θ(bk µ) = � k∈R λk � � a∈An\\{¯a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ˆa} �� θ∈Θ µθφθ(a)us θ(ak) − l(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ak) � + � θ∈Θ µθφθ(¯a)us θ(¯ak) − l(¯a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ¯ak) + � θ∈Θ µθφθ(ˆa)us θ(ˆak) − l(ˆa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ˆak) � + � k /∈R λk � θ∈Θ µθus θ(bk µ) = � k∈R λk � a∈An �� θ∈Θ µθφθ(a)us θ(ak) − l(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ak) � + � k /∈R λk � θ∈Θ µθus θ(bk µ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' showing that the seller’s utility does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, Constraints (15b) relative to ¯a are satisfied since have the form 0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The Constraints (15b) relative to ˆa continue to be satisfied since we add a term l(¯a) on both sides of the inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, all the other Constraint (15b) are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Consider Constraint (15c) relative to a buyer’s type k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It holds � a∈An �� θ∈Θ µθφθ(a)uk θ(ak) + l′(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ak) � − bk = � a∈An\\{¯a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ˆa} �� θ∈Θ µθφθ(a)uk θ(ak) + l(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ak) � + � θ∈Θ µθφθ(¯a)uk θ(¯ak) + � θ∈Θ µθφθ(ˆa)uk θ(ˆak) + l(ˆa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ˆak) + l(¯a) − bk = � a∈An\\{¯a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ˆa} �� θ∈Θ µθφθ(a)uk θ(ak) + l(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ak) � + � θ∈Θ µθφθ(¯a)uk θ(¯ak) + l(¯a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ¯ak) + � θ∈Θ µθφθ(ˆa)uk θ(ˆak) + l(ˆa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ˆak) − bk 36 ARXIV PREPRINT - FEBRUARY 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2023 = � a∈An �� θ∈Θ µθφθ(a)uk θ(ak) + l′(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ak) � − bk ≥ � θ∈Θ µθuk θ(bk µ) Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' we can show that Constraints (15d) continue to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, iteratively applying this procedure we ob- tain a solution with the same value of the optimal protocol and such that for each tuple (a, k) if l(a, ak) > 0 and � θ∈Θ µθφθ(a) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We can convert this solution into an optimal protocol, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', an optimal solution to Problem (6) setting π(a, ak) = l(a,ak) � θ∈Θ µθφθ(a) for each a ∈ An such that � θ∈Θ µθφθ(a) = 0 and k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, we set all the other payments to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It is easy to see that the obtained protocol is a feasible optimal solution to Problem (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 37' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'}