diff --git "a/3tFST4oBgHgl3EQfZDim/content/tmp_files/2301.13790v1.pdf.txt" "b/3tFST4oBgHgl3EQfZDim/content/tmp_files/2301.13790v1.pdf.txt" new file mode 100644--- /dev/null +++ "b/3tFST4oBgHgl3EQfZDim/content/tmp_files/2301.13790v1.pdf.txt" @@ -0,0 +1,4523 @@ +arXiv:2301.13790v1 [cs.GT] 31 Jan 2023 +SELLING INFORMATION WHILE BEING AN INTERESTED PARTY +ARXIV PREPRINT +Matteo Castiglioni +Politecnico di Milano +matteo.castiglioni@polimi.it +Francesco Bacchiocchi +Politecnico di Milano +francesco.bacchiocchi@polimi.it +Alberto Marchesi +Politecnico di Milano +alberto.marchesi@polimi.it +Giulia Romano +Politecnico di Milano +giulia.romano@polimi.it +Nicola Gatti +Politecnico di Milano +nicola.gatti@polimi.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. 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. 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. 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. 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. +We study settings in which buyer’s budget and utilities are determined by a random buyer’s type +that is unknown to the seller. 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. 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. +First, we show that the problem of computing a seller-optimal protocol can be solved in polynomial +time. This result relies on a quadratic formulation of the problem, which we solve by means of a +linear programming relaxation. 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. 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. 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. Such an algorithm runs in polynomial time whenever either the number +of buyer’s actions or that of states of nature is fixed. In order to derive our results, we draw a +connection between our information-selling problem and principal-agent problems with observable +actions. 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. + +ARXIV PREPRINT - FEBRUARY 1, 2023 +1 +Introduction +Nowadays, there is a terrific amount of information being collected on the Web and other online platforms. Such infor- +mation ranges from consumer preferences, e.g., in e-commerce and streaming websites, to credit reports and location +histories. 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. +Very recently, information markets have also received the attention of the algorithmic game theory research commu- +nity. 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., 2019) are now proliferating, only few papers studied +the problem of information selling, with (Babaioff et al., 2012) and (Chen et al., 2020) constituting two notable exam- +ples. +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. Differently from previous works +addressing such a problem (see, e.g., (Chen et al., 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. 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. 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. The realized +state of nature is revealed to the seller, while it remains unknown to the buyer. 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. +We study settings in which buyer’s budget and utilities are determined by a random buyer’s type that is unknown to the +seller. In such settings, in order to optimally sell information, the seller has to commit upfront to a protocol working +as follows. 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. +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. 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. 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. Our protocol extends the one of Chen et al. (2020) by +adding a final payment from the seller to the buyer. 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. Moreover, the addition of payments from the buyer to the seller is also reasonable in many +real-world scenarios. 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. +1.1 +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. We show that +a seller-optimal protocol can be computed in polynomial time. In order to do that, we first formulate the problem of +finding a seller-optimal protocol as a quadratic problem. 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. 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. These results allow us to formulate a suitable linear relaxation of the quadratic problem. A +similar technique has been employed in generalized principal-agent problems (Gan et al., 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. Indeed, Castiglioni et al. (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. 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. As a byproduct, this also shows that, in our setting, the seller’s problem always +admits a maximum. +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. 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. 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. +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. 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. 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. 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. 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. +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. These special settings are of interested on their own, as a similar model has +been recently addressed by Dughmi et al. (2019). We focus on special cases where one of the parameters characterizing +a problem instance is fixed. In particular, we study what happens if we fix the number of buyer’s actions, showing +that the problem admits a PTAS. 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. +Notice that such a loss is exponentially small in 1 +ρ, and, thus, it is negligible even for reasonably large values of ρ. As +shown by Castiglioni et al. (2022a), such an approximation result is tight for hidden-action principal-agent problems. +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. +Table 1: Summary of the results provided in the paper. 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. The approximation guarantees that are shaded in gray can only be obtained by means of a quasi-polynomial-time +algorithm. +general +fixed # actions +fixed # states +fixed # types +Protocols with menus +P +P +P +P +Protocols w/o menus +Buyer w. 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. 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. 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. 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. Finally, we show that, when the number of buyer’s types is fixed, the problem +admits a polynomial-time algorithm. This also implies that the seller’s optimization problem for protocols without +menus always admits a maximum. +We summarize the results provided in this paper in Table 1. All the proofs are in the Appendix. +3 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +1.2 +Related Works +The study of algorithmic ways of selling information to an imperfectly-informed buyer has received some attention +in the past. Babaioff et al. (2012) initiated the study by considering a buyer with an unlimited budget. 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. The main drawback of the approach presented by Babaioff et al. +(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. Chen et al. (2020) complement the results in (Babaioff et al., 2012) +by studying the problem of selling information when both the buyer and the seller are budget-constrained. Moreover, +they also consider a setting in which the buyer’s budget is private, and the seller needs to elicit it in the mechanism. +Chen et al. (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. +The problem of selling information has also been addressed by Bergemann et al. (2018), who study the case of bi- +nary actions and states of nature, characterizing a revenue-maximizing mechanism in such a setting. Furthermore, +Bergemann et al. (2022) extend the analysis to the case in which there are more than two actions and binary states +of nature. In contrast, Liu et al. (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. +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. Dughmi et al. (2019) generalize the classical framework by considering the case in which there are +monetary transfers between the sender and the receiver. Our information-selling setting in which the buyer has limited +liability generalizes the model of Dughmi et al. (2019) by also introducing buyer’s types. +Finally, let us remark that our information-selling problem shares critical features with Bayesian principal-agent prob- +lems (see, e.g., (Castiglioni et al., 2022a; Alon et al., 2021, 2022; Guruganesh et al., 2021; Castiglioni et al., 2022c) +for some references). 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. Such a connection between the +two settings is also demonstrated in terms of results. In particular, notice that Castiglioni et al. (2022a) design bi- +criteria approximation algorithms whose guarantees are similar to those provided in this paper. Moreover, Gan et al. +(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. +2 +Preliminaries +We study the problem faced by an information holder (seller) selling information to a budget-constrained decision +maker (buyer). 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. +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. 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. 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. Specifically, for every state θ ∈ Θ, the sender gets an additional utility contribution determined by a function +us +θ : A → [0, 1]. 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.1 We +let µθ be the probability assigned to state θ ∈ Θ, while λk is the probability of type k ∈ K. +As in (Chen et al., 2020), we assume w.l.o.g. that information revelation happens only once during the seller-buyer +interaction. 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. Formally, φ : Θ → ∆S, where S is a finite set of signals. 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. +2.1 +Protocols with Menus +An information-selling protocol for the seller is defined as follows. The seller first proposes a menu of signaling +schemes to the buyer, with each signaling scheme being assigned with a price. 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. +4 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +scheme and pays its price upfront, before information is revealed.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. This is in contrast with what happens in the protocol introduced by Chen et al. (2020), where there are +no action-dependent money transfers. 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. (2020) where the +seller’s utility function is only determined by how much the buyer pays for acquiring information. Formally, we define +a seller’s protocol as follows: +Definition 1 (Seller’s protocol). 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; +• {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;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.4 +The seller and the buyer interact as follows: (i) the seller commits to a protocol {(φk, pk, πk)}k∈K; (ii) the buyer +selects a signaling scheme φk and pays pk to the seller (with k ∈ K possibly different from their true type); (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; (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); +(v) given the posterior ξs, the buyer selects an action a ∈ A; and (vi) the seller pays πk(s, a) to the buyer. As in the +model by Chen et al. (2020), we assume that the seller is committed to following the protocol, while the buyer is not, +i.e., the buyer is free of leaving the interaction at any point. +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. Formally: +Definition 2 (ǫ-Best-response). Let ǫ ≥ 0. 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 +� +. +We let bk,ǫ +ξs,π ∈ Bk,ǫ +ξs,π be an ǫ-best response played by the buyer. 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.5 +In the following, we will oftentimes work in the space of the distributions over posteriors. 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. +The seller’s goal is to implement an optimal (i.e., utility-maximizing) protocol {(φk, pk, πk)}k∈K. We focus on seller’s +protocols that are incentive compatible (IC) and individually rational (IR).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., 2020). +3Assuming pk ≥ 0 is w.l.o.g., since, intuitively, the seller is never better off paying the buyer before they played any action. +4The assumption that πk(s, a) ≥ 0 is w.l.o.g., 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. +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. +6By a revelation-principle-style argument (see (Shoham and Leyton-Brown, 2008) for some examples), focusing on IC and IR +protocols is w.l.o.g. when looking for an optimal protocol. +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). +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. 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 +� +. +A crucial component of our results is that we can restrict the attention to protocols that are direct and persuasive. 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. +2.2 +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.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. 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. +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. +Formally, this amounts to asking that bk = 0 for all k ∈ K. 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. (2019). Indeed, our model can be seen as a generalization of the one in (Dughmi et al., 2019), +which adds buyer’s private types. 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. (2019) by adding the possibility for the seller to ask the buyer a +payments before information is revealed. +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. Given a protocol without menus +(φ, p, π), only some of the buyer’s types are actually incentivized to participate in the protocol, i.e., all the types whose +corresponding IR constraint is satisfied. 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; +and (ii) the IR constraint is satisfied for a buyer of type k.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). +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. 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). +7From the point of view of Chen et al. (2020), this is equivalent to assuming that there is no type reporting stage. +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. +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). 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). Thus, instead of working with signaling +schemes φ, one can w.l.o.g. work with distributions γ over ∆Θ that are consistent with the probability distribution µ, +i.e., they satisfy the condition in Equation (1). +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. Similarly, we slightly abuse notation in payment functions, by assuming that they are defined over +posteriors rather than signals. 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. +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. 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.t. +(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. +(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.g., (Castiglioni et al., 2022b; Gan et al., 2022)), it is not in general immediate to establish whether the seller’s +optimization problem always admits a maximum or not. Indeed, in the following we show that our problem always +admits a maximum. +As a first step, we prove that we can focus w.l.o.g on protocols which are direct and persuasive. +Lemma 1. 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. +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. 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. 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. +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.e., they get a null payment) if they deviate from the seller’s action +recommendation. Formally: +7 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Lemma 2. 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. +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.l.o.g. on payment functions such that πk(a, a′) = 0 for all a ̸= a′. +We are now ready to introduce an LP with polynomially-many variables and constraints that is a linear relaxation of +Problem (2). 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. Moreover, we encode the terms � +θ∈Θ µθφk +θ(a)πk(a) as single variables lk(a). 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.t. +(3a) +� +a∈A +�� +θ∈Θ +µθφk +θ(a)uk +θ(a) + lk(a) +� +− bk ≥ +� +a∈A +yk,k′,a − bk′ +∀k ∈ K, ∀k′ ∈ K +(3b) +yk,k′,a ≥ +� +θ∈Θ +µθφk′ +θ (a)uk +θ(a) + lk′(a) +∀k ∈ K, ∀k′ ∈ K, ∀a ∈ A +(3c) +yk,k′,a ≥ +� +θ∈Θ +µθφk′ +θ (a)uk +θ(a′) +∀k ∈ K, ∀k′ ∈ K, ∀a ̸= a′ ∈ A +(3d) +� +a∈A +�� +θ∈Θ +µθφk +θ(a)uk +θ(a) + lk(a) +� +− bk ≥ +� +θ∈Θ +µθuk +θ(a′) +∀k ∈ K, ∀a′ ∈ A +(3e) +� +θ∈Θ +µθφk +θ(a)uk +θ(a) + lk(a) ≥ +� +θ∈Θ +µθφk +θ(a)uk +θ(a′) +∀k ∈ K, ∀a ̸= a′ ∈ A +(3f) +� +a∈A +φk +θ(a) = 1 +∀k ∈ K, ∀θ ∈ Θ. +(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. +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). 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. 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. +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. First, we prove a preliminary result: +Lemma 3. The optimal value of LP (3) is at least as large as the supremum in Problem (2). +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. Formally: +Lemma 4. 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). +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.e., action a is recommended with strictly positive probability. Moreover, such a procedure does +not detriment the objective function value and retains the IC and IR conditions. 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. +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. 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. Tanks to Lemma 3, such a protocol is optimal for the +seller. +Theorem 1. There exists a polynomial-time algorithm that computes a protocol with menus that maximizes the seller’s +expected utility. +Theorem 1 also shows as a byproduct that Problem (2) always admits a maximum. +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. (2022). 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. This is because the problem may not admit a maximum, as shown by Castiglioni et al. +(2022b) even in the special case of hidden-action principal-agent problems. Surprisingly, in our information-selling +setting, the linear relaxation can be used to find an (exact) optimal solution. 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. +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. 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. +In Section 4.1, we formally introduce principal-agent problems with observable actions. Then, in Section 4.2, 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 4.3, we provide some preliminary technical results that will be useful +in the following sections. +4.1 +Principal-agent Problem with Observable Actions +We start by formally defining an instance of (Bayesian) observable-action principal-agent problem.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. 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. Each +agent’s type k ∈ K has a set A of actions, with each action having a type-dependent cost ck +a ∈ [0, 1]. The principal is +characterized by a reward ra ∈ [0, 1] for every agent’s action a ∈ A. 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. 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. 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)]. +4.2 +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.e., bk = 0 for all k ∈ K) is strongly related to the problem of finding an optimal (i.e., expected-utility- +maximizing) contract in observable-action principal-agent problems. 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. 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. +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. +9 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Definition 3 (Optimal payment function in a posterior). 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′) +� +. +(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. For the +same reason, we can safely assume that all the buyer’s types satisfy IR constraints. Then, we can state the following +crucial result: +Lemma 5. 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). +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. +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). Thus, by Lemma 5, we can easily prove +the following: +Theorem 2. 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. +While the computational complexity of finding optimal contracts in hidden-action principal-agent problems is well +understood (see, e.g., (Castiglioni et al., 2022a)), to the best of our knowledge, there are no results on problems with +observable actions. 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. Formally: +Theorem 3. In observable-action principal-agent problems, the problem of computing a contract maximizing the +principal’s expected utility is APX-hard. +Then, Theorem 2 immediately gives the following result: +Corollary 1. 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. +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. Such an approximation is +similar to the the bi-criteria guarantees provided by Castiglioni et al. (2022a) for Bayesian hidden-action principal- +agent problems. 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. +Theorem 7 in (Castiglioni et al., 2022a) shows that, for hidden-action problems, such bi-criteria approximations are +tight. 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. +4.3 +Additional Preliminary Technical Results +We conclude the section by recalling two already-known results on hidden-action principal-agent problems. Clearly, +these also hold for the specific case of observable-action principal-agent problems. By Theorem 2, such results can be +easily cast to our information-selling problem. Indeed, we also show that one of them can be strengthen in our setting. +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. 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. Despite their simplicity, +linear contracts provide good approximations with respect to general ones. In particular, the following holds: +Theorem 4 (Essentially Theorem 3 by Castiglioni et al. (2022a)). 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). +Moreover, such a linear contract is defined by a parameter β = 1 − 2−i, for some i ∈ {1, . . ., ⌊1/2ρ⌋}. +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. (2022a) and Lemma 5. Formally: +Corollary 2. 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). +Moreover, such a parameter β is equal to 1 − 2−i for some i ∈ {1, . . . , ⌊1/2ρ⌋}. +Finally, we recall a useful result that establishes a connection between agent’s best responses and approximate best +responses in principal-agent problems. 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. 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. Such a set is made by all the actions a ∈ A such that π(a) − ck +a ≥ maxa′∈A +� +π(a′) − ck +a′ +� +− ǫ. 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. Then: +Theorem 5 (Essentially Proposition A.4 by Dutting et al. (2021)). 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. +Theorem 5 can be easily cast to our setting by means of Lemma 5. Formally: +Corollary 3. 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. +Corollary 3 will be crucial to provide our results in the following sections. +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.e., each buyer’s types k ∈ K has budget bk = 0. As remarked in Section 2.2, such a setting is of interest +on its own, since it is a generalization of the one addressed by Dughmi et al. (2019). 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. We +show how to circumvent the APX-hardness result that we established in Corollary 1, first, in Section 5.1, by fixing the +number of buyer’s actions, and then, in Section 5.2, by fixing the number of states of nature. +In this section, since the buyer has limited liability, we can assume w.l.o.g. that p = 0, so that we can compactly +denote a protocol with a pair (γ, π), rather than with (γ, p, π). +11 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +5.1 +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.e., 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. +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). 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.t. +(5a) +� +ξ∈supp(γ) +γξ ξθ = µθ +∀θ ∈ Θ. +(5b) +Notice that Problem (5) is defined over general distributions over posteriors γ, whose support supp(γ) may be not finite. +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. +We need to introduce a particular class of posteriors, which are called q-uniform posteriors. +Definition 4 (q-Uniform posterior). 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. +In the following, we denote by Ξq ⊆ ∆Θ (for a given q ∈ N>0) the finite set of all the q-uniform posteriors. As it is +easy to check, such a set satisfies |Ξq| ≤ min{dq, qd}. +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). Formally: +Lemma 6. 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.e., π solves Problem (4) for every ξ ∈ Ξq with bk +ξ,π replaced by bk,ǫ +ξ,π. +Furthermore, the signaling scheme γ satisfies: +� +ξ∈Ξq +γξ ξθ = ξ∗ +θ +∀θ ∈ Θ. +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. 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. +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.e., a payment function solving Problem (4) for such a posterior). 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. Thus, by Theorem 3, such a problem is APX-hard in general. 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. 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. Thus, in principle we should start by defining the problem with +a sup rather than a max. However, in Section 6.2, 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. +12 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Lemma 7. 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. +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. +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. 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. +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. By Lemma 6, we can focus on signaling schemes supported over q-uniform posteriors, for a +suitably-defined q ∈ N>0. 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. +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. 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. 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. However, the +overall running time depends exponentially in α > 0, which the seller’s expected utility approximation provided by +the algorithm. This allows us to prove the following result: +Theorem 6. 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. +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. 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. While in general the problem +is APX-hard, an approximately-optimal price function can be computed in polynomial time by applying Corollary 2. +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. This gives the following result: +Theorem 7. 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. +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. +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. 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. +5.2 +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. 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. 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. +Similarly to Section 5.1, 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. 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. +Formally: +13 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Lemma 8. 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. Furthermore, the signaling scheme γ satisfies: +� +ξ∈Ξq +γξ ξθ = ξ∗ +θ +∀θ ∈ Θ. +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. 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. 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. 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. Formally: +Theorem 8. 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. 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. +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. +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. Thus, in such a setting, the seller also decides a price p ∈ R+ for +the signaling scheme proposed to the buyer. +First, we provide a negative result for general instances that is stronger than the one established in Corollary 1. 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. 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. 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. 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.11 +Definition 5 (LINEQ-MA(1−ζ, δ) by Guruswami and Raghavendra (2009)). 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; +• every possible vector x ∈ Qnvar satisfies less than a fraction δ of the equations. +Theorem 9 (Guruswami and Raghavendra (2009)). For all the constants ζ, δ ∈ R which satisfy 0 ≤ δ ≤ 1 − ζ ≤ 1, +the problem LINEQ-MA(1 − ζ, δ) is NP-hard. +11In +the +definition +in +(Guruswami and Raghavendra, +2009), +the +vector +ˆx +can +be +non-binary. +How- +ever, Guruswami and Raghavendra (2009) use a binary vector ˆx in their proof and, thus, their hardness result also holds for +our definition. +14 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Then, Theorem 5 allows us to prove the following hardness result: +Theorem 10. The problem of computing a seller-optimal protocol without menus is APX-hard, even when the number +of buyer’s actions m is fixed. +In the following, we show how to circumvent the hardness result in Theorem 10, by providing, in Section 6.1, a +quasi-polynomial-time bi-criteria approximation algorithm and, in Section 6.2, a polynomial-time (exact) algorithm +working when the number of buyer’s types is fixed. +6.1 +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. Formally, our algorithm guarantees a multiplicative approximation ρ of the optimal utility, by only suffering +an additional 2−Ω(1/ρ) + α additive loss. 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. +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. Our algorithm +works by computing three protocols without menus, each one approximating one of the three terms. Choosing the best +protocol among the three provides the desired approximation guarantees. 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. Such a utility term can be trivially achieved by a protocol that charges no price, discloses no +information, and never pays back the buyer. +• The second term is related to the best seller’s expected utility which can be extracted from a buyer’s action. +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. Thus, the second utility term can be +approximated by using either the algorithm provided in Theorem 7 or that given in Theorem 8.12 +• 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. 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. +Formally, we prove the following main result: +Theorem 11. 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. 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. +6.2 +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. As a byproduct, the existence of such an +algorithm shows that, for protocols without menus, the seller’s optimization problem always admits a maximum. +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. 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. With a slight abuse of notation, we say that +protocols without menus (φ, p, π) as described above are generalized-direct and generalized-persuasive. Formally, we +prove the following result: +Lemma 9. 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. +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. 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. +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. Then, by iterating such a process, we recover a signaling scheme and a payment function employing An as +set of signals . +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. Formally: +Lemma 10. 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. +Finally, equipped with Lemma 9 and Lemma 10, we are ready to provide our polynomial-time algorithm. 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. This can be done in polynomial time since the number of buyer’s types +n is fixed. 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.t. +(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. +(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. This allows us to state our last results: +Theorem 12. 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. +Corollary 4. The problem of computing a seller-optimal protocol without menus always admits a maximum. +References +Paola Alimonti and Viggo Kann. 2000. Some APX-completeness results for cubic graphs. Theoretical Computer +Science 237 (04 2000), 123–134. https://doi.org/10.1016/S0304-3975(98)00158-3 +Tal Alon, Paul Dütting, and Inbal Talgam-Cohen. 2021. Contracts with Private Cost per Unit-of-Effort. In Proceedings +of the 22nd ACM Conference on Economics and Computation. 52–69. +Tal Alon, Paul Dütting, Yingkai Li, and Inbal Talgam-Cohen. 2022. +Bayesian Analysis of Linear Contracts. +arXiv:cs.GT/2211.06850 +Moshe Babaioff, Robert Kleinberg, and Renato Paes Leme. 2012. Optimal mechanisms for selling information. 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Proceedings of the 22nd ACM +Conference on Economics and Computation (2021). +Yoav Shoham and Kevin Leyton-Brown. 2008. Multiagent systems: Algorithmic, game-theoretic, and logical founda- +tions. Cambridge University Press. +17 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +A +Proofs Omitted from Section 3 +Lemma 1. 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. +Proof. 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.e., b¯k +ξs1 = b¯k +ξs2 = ¯a. +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. Formally we define: ¯φ¯k +θ(¯s) = φk +θ(s1) + φk +θ(s2) for each θ ∈ Θ. 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.e., we leave these components of +{ +�¯φk, ¯pk, ¯πk +� +}k∈K equal to the one in { +� +φk, pk, πk +� +}k∈K. +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. 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. +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. 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. +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. Due to +that, we can easily conclude that the new protocol preserves the IC and the IR constraints. 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. 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. +Lemma 2. 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. +Proof. 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. First, we show that by taking ˜πk(a, a′) = 0 for each +a ̸= a′ ∈ A and k ∈ K, the signaling schemes φk remain persuasive. 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′). +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. 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. Hence, the IC constraints are satisfied. 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. +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. 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. 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 +� +. +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. +Lemma 3. The optimal value of LP (3) is at least as large as the supremum in Problem (2). +Proof. 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. This is sufficient to prove the statement. +By Lemma 1 and Lemma 2, we focus without loss of generality on protocols { +� +φk, pk, πk +� +}k∈K that are direct and +persuasive. 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. Moreover, we let ¯φ = φ. It is easy to verify that the solution (¯φ, ¯l, ¯y) results feasible +for LP (3). +Lemma 4. 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). +Proof. 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. 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. 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). +It is easy to check that the solution +(¯φ, ¯l, ¯y)achieves the same seller’s expected utility while satisfying the constrains. 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. +19 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +To recover a feasible protocol we just need to set payments as follows. 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)). Otherwise, we set ˜πk(a) = 0. Notice that the ratio +˜lk(a)/(� +θ∈Θ µθ ˜φk +θ(a)) is always well defined. 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) . This concludes +the proof. +Theorem 1. There exists a polynomial-time algorithm that computes a protocol with menus that maximizes the seller’s +expected utility. +Proof. The algorithm solves LP 3. By Lemma 3, this solution has value greater or equal to the supremum of Program 2. +Then, exploiting Lemma 4, we can recover in polynomial-time a protocol with at least the same utility, i.e., an optimal +one. +B +Proofs Omitted from Section 4 +Lemma 5. 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). +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. +Proof. We start proving the first part of the statement. 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). To prove +the equivalence between the two settings, we first show that the set of best-responses for the two problems coincides. +Indeed, given a payment function π and a type k ∈ K, let a ∈ Bk +ξ,π. 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. 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.e., Bk +π = Bk +ξ,π. 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. This proves the first part of the statement. +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. 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. Following a analysis similar to the first +part of the proof, we can show that the two problems are equivalent. This concludes the proof. +20 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Theorem 3. In observable-action principal-agent problems, the problem of computing a contract maximizing the +principal’s expected utility is APX-hard. +Proof. We reduce from vertex cover in cubic graphs. 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). Let +η = ε/7. 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. +Consider an instance of vertex cover (V, E) with nodes V and edges E. Let ρ = |V | and ℓ = |E|. 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. Similarly, given an edge e ∈ E, +let V (e) be the set of vertexes v such that e is an edge with extreme v. We build an instance of the principal-agent +problem with observable actions as follows. For each vertex v ∈ V , there exists an agent’s type kv, while for each +e ∈ E, there exists a type ke. For each vertex v ∈ V , there exists an action av and an additional action a−. The cost +of a type ke, e ∈ E, is cke +a− = 0, cke +av = 1 +2 if e ∈ E(v) and 1 otherwise. The cost of a type kv, v ∈ V , is ckv +av = 0, and +1 otherwise. Finally, the principal’s utility is equal to 1 if the action is in {av}v∈V , while is equal to 0 otherwise, i.e., +us +av = 1 for each v ∈ V and us +a− = 0. All the types are equally probable, i.e., λk = +1 +ρ+ℓ for each k ∈ K. +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 +. Consider +the payment function such that π(av) = 1 +2 if v ∈ V ⋆ and 0 otherwise. A type kv with v /∈ V ⋆ plays the action av and +receives a payment of 0. A type kv with v ∈ V ⋆ plays the action av and receives a payment of 1 +2. A type ke plays an +action av such that e ∈ E(v) (this action exists by construction) and receives a payment of 1 +2. It is easy to see that the +expected seller’s utility is (ρ−ν)+ ν +2 + ℓ +2 +ρ+ℓ +. +Suppose that there exists an algorithm that provides a 1 − η approximation. This implies that the algorithm returns a +solution, i.e., a payment function π, with value at least (1 − η) ρ− k +2 + ℓ +2 +ρ+ℓ +. We show how to exploit the payment function +π to build a vertex cover of size at most (1 + ε)ν in polynomial time. In particular, given π we recover a vertex +cover ¯V of the desired size as follows. 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. Intuitively, payments are useful only to change the best +response of a type ke, e ∈ E, from a− to av, v ∈ V . To do so, it is sufficient a payment of 1 +2. Then, let ¯E be the set +of edges e ∈ E such that the best response of ke is a−, i.e., ¯E = {e ∈ E : bke +π = a−}. Consider a edge e ∈ ¯E and +a vertex v ∈ V (e). Since e ∈ ¯E the payment π(av) = 0 and no type ke plays action av. 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. 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. 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. This implies that the principal’s +utility does not decrease with this procedure. 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 . Then, let ¯V be the set of vertexes with at least +one agent of type ke, e ∈ E that plays this action, i.e., ¯V = {v ∈ V : bke +π = av, e ∈ E}. We show that ¯V is an vertex +cover of size at most (1 + ǫ)ν, concluding the proof. Notice that we can set payment π(av) = 0 for each v ∈ V \ ¯V +without decreasing the seller’s utility. 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. 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. 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. This concludes the proof. +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. +21 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Corollary 2. 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). +Moreover, such a parameter β is equal to 1 − 2−i for some i ∈ {1, . . . , ⌊1/2ρ⌋}. +Proof. As a first step, we notice that even if Castiglioni et al. (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.e., � +k∈K λk maxa∈A[ra −ck +a]. To prove this result, it +is sufficient to follow all the steps of Theorem 3 of (Castiglioni et al., 2022a) except for the one in which Observation 1 +is employed to upperboud the value of the optimal contract with the social welfare. Finally, we can modify this result +to hold in our setting exploiting Lemma 5. +C +Proofs Omitted from Section 5 +Lemma 6. 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.e., π solves Problem (4) for every ξ ∈ Ξq with bk +ξ,π replaced by bk,ǫ +ξ,π. +Furthermore, the signaling scheme γ satisfies: +� +ξ∈Ξq +γξ ξθ = ξ∗ +θ +∀θ ∈ Θ. +Proof. Let ˜ξ ∈ Ξq be the empirical mean of q i.i.d. samples drawn according to ξ∗ ∈ ∆Θ, where each θ ∈ Θ +has probability ξ∗ +θ of being sampled. Therefore, ˜ξ ∈ Ξq is a random vector supported on q-uniform posteriors with +expectation ξ∗ ∈ ∆Θ. Moreover, let γ ∈ ∆Ξq be a probability distribution such as, for each ξ ∈ Ξq, it holds γξ := +Pr(˜ξ = ξ). 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. +(7) +Then, for each ξ ∈ Ξq,ǫ, we have that Bk +ξ∗,π ⊆ Bk,ǫ +ξ,π. 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. (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. +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). 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. +(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. +(By Equation (8)) +Let ¯ξ be a d-dimensional vector defined as ¯αθ := � +ξ∈Ξq\Ξq,ǫ γξξθ. By definition and for the previous result we have: +� +θ∈Θ ¯ξθ ≤ α. Finally, we can show: +� +ξ∈Ξq +γξ +� +θ∈Θ +ξθus +θ(bk,ǫ +ξ,π′) − π′(ξ, bk,ǫ +ξ,π′) +≥ +� +ξ∈Ξq +γξ +� +θ∈Θ +ξθus +θ(bk,ǫ +ξ,π) − π(ξ, bk,ǫ +ξ,π) +≥ +� +ξ∈Ξq,ǫ +γξ +� +θ∈Θ +ξθus +θ(bk,ǫ +ξ,π) − π(ξ, bk,ǫ +ξ,π) +≥ +� +ξ∈Ξq,ǫ +γξ +� +θ∈Θ +ξθus +θ(bk +ξ∗,π) − π(ξ∗, bk +ξ∗,π) +(Bk +ξ∗,π ⊆ Bk,ǫ +ξ,π for each ξ ∈ Ξq,ǫ) += +� +θ∈Θ +� +us +θ(bk +ξ∗,π) − π(ξ∗, bk +ξ∗,π) +�� � +ξ∈Ξq,ǫ +γξξθ +� += +� +θ∈Θ +� +us +θ(bk +ξ∗,π) − π(ξ∗, bk +ξ∗,π) +�� � +ξ∈Ξq +γξξθ − ¯ξθ +� +(By definition of ¯α) += +� +θ∈Θ +� +ξθus +θ(bk +ξ∗,π) − π(ξ∗, bk +ξ∗,π) +�� � +ξ∈Ξq +γξξθ +� +− +� +θ∈Θ +� +us +θ(bk +ξ∗,π) − π(ξ∗, bk +ξ∗,π) +� +¯ξθ +≥ +� +θ∈Θ +� +us +θ(bk +ξ∗,π) − π(ξ∗, bk +ξ∗,π) +�� � +ξ∈Ξq +γξξθ +� +− +� +θ∈Θ +¯ξθ +(Utilities in [0, 1]) +≥ +� +θ∈Θ +ξ∗ +θus +θ(bk +ξ∗,π) − π(ξ∗, bk +ξ∗,π) − α. +Finally, by definition of γ, we have that, for each θ ∈ Θ: +� +ξ∈Ξq +γξξθ = ξ∗ +θ. +This concludes the proof. +Lemma 7. 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. +Proof. 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. 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.t. +� +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 . +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. +Hence, the total number of vertexes defining all the regions Πa, a ∈ A|K|, is at most +�nm2+m +m +� += O((nm2 + m)m). +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∗. Moreover, there are overall +�� � +a∈A|K| V (Πa) +�� = +O((km2 + m)m) vertexes, where V (·) denotes the set of vertexes of the polytope. 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. +Theorem 6. 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. +Proof. Given two arbitrary constants α, ǫ > 0 we let (γ, π) be an optimal protocol. We show that an (α+2√ǫ)-optimal +protocol (γ∗, π∗) can be computed in polynomial time. 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 +. 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. +Moreover, let π∗ : ∆Θ × A → R+ be the optimal payment function in each ˜ξ ∈ Ξq. We show that the protocol +(γ∗, π∗, 0) is (α + 2√ǫ)-optimal. Let π′′ : ∆Θ × A → R+ be the optimal payment function in each ξ ∈ Ξq when the +buyer is playing an ǫ-best response, i.e., +π′′(ξ, ·) ∈ arg max +˜π(ξ,·) +� +k∈K +λk[ +� +θ +ξθus +θ(bk,ǫ +ξ,˜π) − ˜π(ξ)]. +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. Then, we have: +� +˜ξ∈Ξq +γ∗ +˜ξ +� � +θ∈Θ +˜ξθus +θ(bk +˜ξ,π∗) − π∗(˜ξ, bk +˜ξ,π∗) +� +≥ +� +˜ξ∈Ξq +γ∗ +˜ξ +� � +θ∈Θ +˜ξθus +θ(bk +˜ξ,π′) − π′(˜ξ, bk +˜ξ,π′) +� +(Optimality of π∗) +≥ +� +˜ξ∈Ξq +γ∗ +˜ξ +� � +θ∈Θ +˜ξθus +θ(bk,ǫ +˜ξ,π′′) − π′′(˜ξ, bk,ǫ +˜ξ,π′′) +� +− 2√ǫ +(By Proposition 3) +≥ +� +ξ∈supp(γ) +γξ +� � +˜ξ∈Ξq +γξ +˜ξ +� � +θ∈Θ +˜ξθus +θ(bk,ǫ +˜ξ,π′′) − π′′(˜ξ, bk,ǫ +˜ξ,π′′) +�� +− 2√ǫ +(By defintion of γ∗) +≥ +� +ξ∈supp(γ) +γξ +� � +θ∈Θ +ξθus +θ(bk +ξ,π) − π(ξ, bk +ξ,π) +� +− α − 2√ǫ +(By Lemma 6). +Notice that the optimal payment π∗ : ∆Θ × A → R+ in each ξ ∈ Ξq can be computed in polynomial time employing +Lemma 7. 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.t. +24 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +� +ξ∈supp(γ) +γξξθ = µθ +∀θ ∈ Θ. +Note that since |Ξq| = O(dq), all the payment function π∗ : Ξq × A → R+ can be precomputed in polynomial time. +Moreover, the LP has polynomially many variables and constraints and can be solved efficiently. Finally, the solution +returned by the LP is α + 2√ǫ-optimal. This concludes the proof. +Theorem 7. 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. +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. +Proof. The proof follows the same steps of Theorem 6. However, it relies on Theorem 4 instead of Lemma 7 to +compute an approximate payment function for all q-uniform posteriors. +Lemma 8. 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. Furthermore, the signaling scheme γ satisfies: +� +ξ∈Ξq +γξ ξθ = ξ∗ +θ +∀θ ∈ Θ. +Proof. We define a payment function π′ : ∆Θ ×A → R+ as follows: π′(ξ, a) = π(ξ∗, a) for each a ∈ A and ξ ∈ ∆Θ. +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. Notice +that if q ≥ 18d +α2 , it holds ξ∗ ∈ co(Ξ(ξ∗)). 14 +We show that for each ξ ∈ Iα(ξ∗), it holds bk +ξ∗,π′ ∈ Bk,ǫ +ξ,π′, where ǫ := α2/9. 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. This shows that bk +ξ∗,π′ ∈ Bk,ǫ +ξ,π′. +Let π∗ : ∆Θ × A → R+ be the payment function prescribed by Proposition 3. 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. +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. +This shows that the expected seller’s utility decreases of at most 3√ǫ when we consider sufficiently close posteriors. +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∗)). We show now that such a decomposition decreases the +expected seller’s utility only by the desired amount. 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. +Since 3√ǫ ≤ α, this concludes the proof. +Theorem 8. 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. 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. +Proof. Given a constant α > 0 we let (γ, π) be an optimal protocol. 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). Moreover, the payment function π∗ is a linear function with parameter β ∈ {1 − 2−i}i∈{i,...,⌊ρ/2⌋}. 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 . 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. +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. +Let π′ be the optimal payment function. We show that the protocol (γ∗, π∗) achieves the desired approximation. +Formally: +� +˜ξ∈Ξq +γ∗ +˜ξ +� � +θ∈Θ +˜ξθus +θ(bk +˜ξ,π∗) − π∗(˜ξ, bk +˜ξ,π∗) +� +26 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +≥ ρ + + � +˜ξ∈Ξq +γ∗ +˜ξ +� � +θ∈Θ +˜ξθus +θ(bk +˜ξ,π′) − π′(˜ξ, bk +˜ξ,π′) +� + + − 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). +This implies that since (γ∗, π∗) is feasible for the following LP, it has value at least ρOPT − 2Ω(1/ρ) − α. +max +γ≥0 +� +k∈K +λk +� +ξ∈Ξq +γξ +� +θ∈Θ +ξθus +θ(bk +ξ,π∗) − π∗(ξ, bk +ξ,π∗) s.t. +� +ξ∈supp(γ) +γξξθ = µθ +∀θ ∈ Θ. +Hence, to find the desired approximation it is sufficient to compute π∗ in each q-uniform posterior and solve the LP. +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. +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. +D +Proofs Omitted from Section 6 +Theorem 10. The problem of computing a seller-optimal protocol without menus is APX-hard, even when the number +of buyer’s actions m is fixed. +Proof. 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; +• 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. +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. 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. +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. It is easy to see +that the normalization preserves the number of satisfiable equations. 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. For every variable i ∈ [nvar], we +define a state of nature θi ∈ Θ. Moreover, we introduce three additional states θ0, θ1, θ2 ∈ Θ. 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). 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 . Moreover, we define an additional type k⋆. All the types +k ∈ K have budget bk = ν/2, where ν will be defined in the following. The buyer has 9 actions available, namely +A := {a0, a1, a2, a3, a4, a5, a6, a7, a8}. Then, we define the utilities of the players, where the utility is 0 when not +specified. 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. +• u +k1 +j +θ1 (a3) = 1 +2 + 2ν, +For each k2 +j , j ∈ [neq], the utilities are: +• u +k2 +j +θi (a0) = 1 +2 − ¯Aji + ¯cj for each i ∈ [nvar], +• u +k2 +j +θi (a7) = 1 +2 for each i ∈ [nvar] +• u +k2 +j +θ0 (a0) = 1 +2 + ¯cj, +• u +k2 +j +θ1 (a3) = 1 +2 + 2ν, +For each type k3 +j , j ∈ [neq] the utilities are: +• u +k3 +j +θi (a0) = 1 +2 + ¯Aji − ¯cj for each i ∈ [nvar], +• u +k3 +j +θi (a7) = 1 +2 for each i ∈ [nvar] +• u +k3 +j +θ0 (a1) = 1 +2 − ¯cj, +• u +k3 +j +θ1 (a3) = 1 +2 + 2ν, +For each type k4 +j , 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, +• ukj +θi (a7) = 1 +2 for each i ∈ [nvar], +Finally, the utilities of type k⋆ are: +• uk⋆ +θ1 (a6) = 1, +• uk⋆ +θ (a1) = 1 for each θ ∈ Θ. +Moreover, we let uk +θ(a8) = 1 +2 for every k ∈ K and θ ∈ Θ. 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. +We recall that the utility is 0 when not defined explicitly. +Completeness. +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. Let X1 ⊆ [nvar] be the set of variables i ∈ [nvar] with ˆxi = 1, while X0 := [nvar] \ X1. 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. Let us +consider an (indirect) signaling scheme φ : Θ → ∆S where the set of signals is S := {s1, s2, s3}. Let q := nvar(nvar−1) +τ−|X1| . +For each i ∈ [nvar], let φθi(s1) = q and φθi(s2) = 1 − q if i ∈ X1, while φθi(s2) = 1 otherwise. Moreover, let +φθ0(s1) = 1, φθ1(s3) = 1 and φθ2(s2) = 1. Then, all the other probabilities φθ(s) are set to 0. It is easy to see that +the signaling scheme is feasible. Moreover, we set the price p = ν/2. Finally, we set π(s3, a6) = 2ν and all the other +payments π(s, a) = 0. +Now, we compute the expected seller’s utility due of each type of buyer. +• 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. +Moreover, in the other posteriors ξs1 and ξs2 the seller’s utility is at least 0. +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. 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. +• Consider a buyer k1 +j , j ∈ [neq], such that the j-th equality is satisfied by the vector ˆx. Now, let us take +the buyer’s posterior ξs1 ∈ ∆Θ induced by the signal s1. Let h := +q +n2var +� +i∈X1 +q +n2var + nvar−1 +nvar +. 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��. The buyer of type kj ∈ K experiences a utility of � +θ∈Θ ξs1 +θ ukj +θ (a0) = 1 +2 +by playing action a0. Instead, 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, +where the second to last equality holds since h = 1 +τ (by definition of h and q), while the last equality follows +from the fact that the j-th equation is satisfied, and, thus, 1 +τ +� +i∈X1 ¯Aji = ¯cj (recall that ¯xi = +1 +τ for all +i ∈ X1). Using similar arguments, we can write � +θ∈Θ ξs1 +θ ukj +θ (a2) = 1 +2. Moreover, all the other actions have +utility 0. Hence, the buyer plays a0 in the posterior ξs1. In posterior ξs2 induced by signal s2, the utility of +each action different from a8 is strictly smaller than 1 +2. 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. Hence the expected utility +of the buyer is 3 +4 +1 +2 + 1 +4( 1 +2 + 2ν) − p = 1 +2. 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. Hence, when the buyer’s type is k1 +j +the expected seller’s utility is νµθ0 + ν/2. +• Consider a buyer k2 +j or k3 +j , j ∈ neq such that the j-th equality is satisfied. 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. +In posterior ξs3, the optimal action is a3. Hence, the expected buyer’s utility is 3 +4 +1 +2 + 1 +4( 1 +2 + 2ν) − p = 1 +2. +Hence, the protocol is IR for the buyer and provides expected seller’s utility at least 2νµθ0 + ν/2. +• Consider a buyer k4 +j , j ∈ [neq] such that the j-th equality is satisfied. The buyer has an utility similar to k1 +j +and plays the same best responses. Hence, it is indifferent in participating or not participating to the protocol. +29 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +We assume that they brake ties in favor of the seller and does not accept. She plays action a5 and the expected +seller’s utility is 2ν. +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. 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. 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). 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. Finally, the maximum +expected utility that the seller can extract from a buyer of type k4 +j , j ∈ [neq], is 1 +82ν. +Using the previous upperbounds, we can bound the component of the utility due to each set of types. 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. To see that, consider for instance the +types k1 +j , j ∈ [neq]. 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. 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]. +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. Suppose by contradiction that there is no posterior in +which a t fraction of k1 +j , j ∈ [neq], plays a0. 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. Moreover, the seller’s utility minus payment is greater than 0 in a posterior only +if the agent plays a0. 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. 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. A similar argument holds for the other types. 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. Notice that |Q| ≥ 1 − 3(1 − t) and for t large enough |Q| > δ. +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. 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). For every buyer’s type k1 +j ∈ K, it holds � +θ∈Θ ξθukj +θ (a0) = 1 +2. 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. +Similarly, it holds: +� +θ∈Θ +ξθu +k1 +j +θ (a2) = 1 +2 − ¯cj + +� +i∈[nvar] +ξθi ¯Aji. +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. +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. +Since we +30 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +have that t types k1 +j play a0, this implies that π(ξ, a0) > 0. However, at the same time we have that the buy- +ers of type k2 +j and k3 +j plays action a7. +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). +Recall that this buyer must play a7. +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). 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). A similar argument +holds for the buyer k3 +j∗ if the second inequality is satisfied. 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. 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. +We conclude the proof showing that the utility of this buyer’s type is too small, reaching a contradiction. First, +notice that the seller must induce a posterior with ξθ1 ≥ +3 +4 with probability at least 1 +8. In all the other posteriors +the seller’s utility from type k∗ is 0. However, it must hold that the utility from type k⋆ is at least +1 +64 for ν small +enough. 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. Now consider the type k4 +j∗ that is IR. In a posterior ξ with ξθ1 ≥ +3 +4, the seller’s utility +when the type is k4 +j∗ is at most 0. 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. 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. 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. Thus, we reach a contradiction. +Theorem 11. 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. 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. +Proof. Let (φ, p, π) be an optimal protocol. 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). +Given a signal s ∈ S and a type k ∈ K, let bk +ξs ∈ arg maxa∈A +� +θ∈Θ µθφθ(s)uk +θ(a). Intuitively, bk +ξs is an opti- +mal action for the buyer without considering the payment function. 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); +(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) +; +(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) +� +� +. +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. +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. We will show that this will implies that at least one +protocol provides a good approximation of the overall seller’s utility, i.e., of U1 + U2 + U3. +Approximate U1. +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. +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, . . . , ⌊ 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. +Now, we need a protocol (φ2, p2, π2) that approximate the utility obtained by the optimal protocol that uses only linear +payment functions. 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. 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. Hence, protocol (φ2, p2, π2) can be computed in time poly(min{Id, Ilog(m)}). +Notice that both the algorithms returns a protocol such that p = 0 and hence p2 = 0. Then, we can show that the +protocol (φ2, p2, π2) has seller’s utility +� +k∈K +λk +�� +s∈S +� +θ +µθφθ(s) +� +us +θ(bk +ξs,π2) − π2(s, bk +ξs,π2) +� +� +≥ +� +k∈K +λk +�� +s∈S +� +θ +µθφθ(s) +� +us +θ(bk +ξs,π′) − π′(s, bk +ξs,π′ +� +� +− α += +� +k∈K +λk +�� +s∈S +�� +θ +µθφθ(s) +� � +θ +ξs +θ +� +us +θ(bk +ξs,π′) − π′(s, bk +ξs,π′ +� +� +− α += +� +s∈S +�� +θ +µθφθ(s) +� � +k∈K +λk +�� +θ +ξs +θ +� +us +θ(bk +ξs,π′) − π′(s, bk +ξs,π′ +� +� +− α +≥ +� +s∈S +�� +θ +µθφθ(s) +� � +ρ +� +k∈R +λk +� +θ +ξs +θ +� +us +θ(bk +ξs,π) + uk +θ(bk +ξs,π) − uk +θ(bk +ξs) +� +− 2−Ω(1/ρ) +� +− α += +� +s∈S +�� +θ +µθφθ(s) +�  +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) − α +32 + +ARXIV PREPRINT - FEBRUARY 1, 2023 += ρ +� +k∈Rφ,p,π +λk +� +s∈S +�� +θ +µθφθ(s) +� �� +θ +ξs +θ +� +us +θ(bk +ξs,π) + uk +θ(bk +ξs,π) − uk +θ(bk +ξs) +� +� +− 2−Ω(1/ρ) − α += ρ +� +k∈Rφ,p,π +λk +� +s∈S +� +θ +µθφθ(s) +� +us +θ(bk +ξs,π) + uk +θ(bk +ξs,π) − uk +θ(bk +ξs) +� +− 2−Ω(1/ρ) − α, +where the first inequality holds since π′ employs linear payments functions and (φ2, p2, π2) has an additive loss α +w.r.t. any protocol that employs linear payments functions, while the second inequality comes from Equation (11). +Approximate U3. +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. For each k ∈ Rφ,p,π, by the definition of IR it holds +� +s∈S +� +θ∈Θ +µθφθ(s)[π(s, bk +ξs,π) + uk +θ(bk +ξs,π)] − p ≥ +� +θ +µθuk +θ(bk +µ), +(12) +Hence, +p − +� +s∈S +� +θ∈Θ +µθφθ(s) +� +π(s, bk +ξs,π) + uk +θ(bk +ξs,π) − uk +θ(bk +ξs) +� += p − +� +s∈S +� +θ∈Θ +µθφθ(s) +� +π(s, bk +ξs,π) + uk +θ(bk +ξs,π) +� ++ +� +s∈S +� +θ∈Θ +µθφθ(s)uk +θ(bk +ξs) +≤ p − +� +s∈S +� +θ∈Θ +µθφθ(s) +� +π(s, bk +ξs,π) + uk +θ(bk +ξs,π) +� ++ +� +s∈S +� +θ∈Θ +µθφθ(s)uk +θ(bk +θ) += p − +� +s∈S +� +θ∈Θ +µθφθ(s) +� +π(s, bk +ξs,π) + uk +θ(bk +ξs,π) +� ++ +� +θ∈Θ +µθuk +θ(bk +θ) +≤ − +� +θ∈Θ +µθuk +θ(bk +θ) + +� +θ∈Θ +µθuk +θ(bk +θ) +≤ δk, +where the first inequality follows by the optimality of action bk +θ in state θ, and the second one by Equation (12). +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. +Let Pζ := {2−i}i∈{1,...,⌊1/ζ⌋} ∪ {0}, and for each k ∈ K let pk be the greatest p ∈ Pζ such that p ≤ δk. 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⌋. +Let p∗ = argmaxp∈Pζ p � +k∈K:pk=p λk. 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. We show that this protocol satisfies the +IR constraint for all the players such that pk = p∗. 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. +(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∗}. +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 (??). 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). +Wrapping up. +Let i = arg maxj∈{1,2,3} Uj and OPT be the seller’s utility with the optimal protocol (φ, p, π). Then, +since U1+U2+U3 = OPT, we have that Ui ≥ 1 +3OPT. 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. 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. This concludes the proof. +Lemma 9. 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. +Proof. 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. 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. 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)). 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∗) +� +. +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∗) +� +. +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. As a consequence, the +two protocols achieve the same seller’s expected utility. +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. +Lemma 10. 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. +34 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Proof. Let (φ, π, p) be a protocol. 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. Let +ˆk ∈ arg +min +k∈Rφ,π,p:bk≥p{bk}. +We observe that all the buyer’s types k ∈ Rφ,π,p have enough budget to participate in the protocol,i.e., bk ≥ bˆk. +Furthermore, we define ˆπ(s, a) = π(s, a) + bˆk − p for each s ∈ S and a ∈ A. +Then, we show that the set of types Rφ,π,p = Rφ,ˆπ,ˆp. 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. Similarly, we can prove that each buyer’s type k /∈ Rφ,ˆπ,ˆp does not belong to Rφ,π,p. It +follows that Rφ,π,p = Rφ,ˆπ,ˆp. +Finally, we can show that the seller’s utility results equal to the one in (φ, π, p). 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. +Theorem 12. 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. +Proof. 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. 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. Furthermore, we can focus on direct protocols by Lemma 9. 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). 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. Then, we obtain the following LP. +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. +(15e) +Hence, once we fix bk and R, LP (15) returns a solution that has the same value of the optimal protocol. +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. Notice that, given a price +p, the IR constraint can be satisfied only the buyer’s type k ∈ K with bk ≥ p. Then, we solve LP (15). Finally, we +return the solution with highest value. As we show in the first part of the proof, this solution has the same value of the +optimal protocol. Moreover, it is easy to check that the overall procedure requires to solve O(n2n) LPs, showing that +the algorithm runs in polynomial time. +To conclude the proof, we need to show how to modify the solution of LP 15 to obtain a protocol, i.e., a solution to +Problem (6), with at least the same value. To do so, we exploit a similar approach to the one presented in Section 3. +Let (φ, l) be the solution to LP (15) returned by the algorithm. Suppose that there exists a couple (¯a, ¯k) such that +l(¯a, a¯k) > 0 and � +θ∈Θ µθφθ(¯a) = 0. We show how to obtain a solution such that l(¯a, a) = 0 for each a ∈ A. Notice +that by Constraint (15b), it holds l(¯a, ¯ak) ≥ l(¯a, a) for each k ∈ K, a ∈ A. This implies that l(¯a, ¯ak) = l(¯a, ¯ak′) for +each k ̸= k′. We denote this value with l(¯a). Let ˆa ∈ An be any signal such that � +θ∈Θ µθφθ(ˆa) > 0. Consider a +assignment (φ, l′) to the variables such that +• l′(¯a, a) = 0 for each a ∈ A; +• l′(ˆa, a) = l(ˆa, a) + l(¯a) for each a ∈ A; +• l′(a) = l(a) for each a /∈ {¯a, ˆa}. +We show that this solution is feasible to LP (15) and has the same objective value of (φ, l). Indeed, it holds +� +k∈R +λk +� +a∈An +�� +θ∈Θ +µθφθ(a)us +θ(ak) − l′(a, ak) +� ++ +� +k /∈R +λk +� +θ∈Θ +µθus +θ(bk +µ) += +� +k∈R +λk +� +� +a∈An\{¯a,ˆa} +�� +θ∈Θ +µθφθ(a)us +θ(ak) − l′(a, ak) +� ++ +� +θ∈Θ +µθφθ(¯a)us +θ(¯ak) ++ +� +θ∈Θ +µθφθ(ˆa)us +θ(ˆak) − (l(ˆa, ˆak) �� l(¯a)) +� ++ +� +k /∈R +λk +� +θ∈Θ +µθus +θ(bk +µ) += +� +k∈R +λk +� +� +a∈An\{¯a,ˆa} +�� +θ∈Θ +µθφθ(a)us +θ(ak) − l(a, ak) +� ++ +� +θ∈Θ +µθφθ(¯a)us +θ(¯ak) − l(¯a, ¯ak) ++ +� +θ∈Θ +µθφθ(ˆa)us +θ(ˆak) − l(ˆa, ˆak) +� ++ +� +k /∈R +λk +� +θ∈Θ +µθus +θ(bk +µ) += +� +k∈R +λk +� +a∈An +�� +θ∈Θ +µθφθ(a)us +θ(ak) − l(a, ak) +� ++ +� +k /∈R +λk +� +θ∈Θ +µθus +θ(bk +µ), +showing that the seller’s utility does not change. Moreover, Constraints (15b) relative to ¯a are satisfied since have the +form 0 ≥ 0. The Constraints (15b) relative to ˆa continue to be satisfied since we add a term l(¯a) on both sides of the +inequality. Finally, all the other Constraint (15b) are unchanged. Consider Constraint (15c) relative to a buyer’s type +k ∈ K. It holds +� +a∈An +�� +θ∈Θ +µθφθ(a)uk +θ(ak) + l′(a, ak) +� +− bk += +� +a∈An\{¯a,ˆa} +�� +θ∈Θ +µθφθ(a)uk +θ(ak) + l(a, ak) +� ++ +� +θ∈Θ +µθφθ(¯a)uk +θ(¯ak) ++ +� +θ∈Θ +µθφθ(ˆa)uk +θ(ˆak) + l(ˆa, ˆak) + l(¯a) − bk += +� +a∈An\{¯a,ˆa} +�� +θ∈Θ +µθφθ(a)uk +θ(ak) + l(a, ak) +� ++ +� +θ∈Θ +µθφθ(¯a)uk +θ(¯ak) ++ l(¯a, ¯ak) + +� +θ∈Θ +µθφθ(ˆa)uk +θ(ˆak) + l(ˆa, ˆak) − bk +36 + +ARXIV PREPRINT - FEBRUARY 1, 2023 += +� +a∈An +�� +θ∈Θ +µθφθ(a)uk +θ(ak) + l′(a, ak) +� +− bk +≥ +� +θ∈Θ +µθuk +θ(bk +µ) +Similarly, we can show that Constraints (15d) continue to hold. 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. We can convert this solution into an optimal protocol, i.e., 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. Moreover, we set all +the other payments to 0. It is easy to see that the obtained protocol is a feasible optimal solution to Problem (6). This +concludes the proof. +37 +