1 Introduction
In posted price auctions, the seller tries to sell an item by proposing takeitorleaveit prices to buyers arriving sequentially. Each buyer has to choose between declining the offer—without having the possibility of coming back—or accepting it, thus ending the auction. Nowadays, posted pricing is the most used selling format in ecommerce (Einav et al., 2018), whose sales reach over $4 trillion in 2020 (eMarketer, 2021). Posted price auctions are ubiquitous in settings such as, for example, online travel agencies (e.g., Expedia), accommodation websites (e.g., Booking.com), and retail platforms (e.g., Amazon and eBay). As a result, growing attention has been devoted to their analysis, both in economics (Seifert, 2006) and in computer science (Chawla et al., 2010; Babaioff et al., 2015, 2017; Adamczyk et al., 2017; Correa et al., 2017)
, within AI and machine learning in particular
(Kleinberg and Leighton, 2003; Shah et al., 2019; Romano et al., 2021).We study Bayesian posted price auctions, where the buyers’ valuations for the item depend on a random state of nature, which is known to the seller only. By applying the Bayesian persuasion framework (Kamenica and Gentzkow, 2011), we consider the case in which the seller (sender) can send signals to the buyers (receivers) so as to disclose information about the state. Thus, in a Bayesian auction, the seller does not only have to decide price proposals for the buyers, but also how to partially disclose information about the state so as to maximize revenue. Our model finds application in several realworld scenarios. For instance, in an ecommerce platform, the state of nature may reflect the condition (or quality) of the item being sold and/or some of its features. These are known to the seller only since the buyers cannot see the item given that the auction is carried out on the web.
Original Contributions.
We study the problem of maximizing seller’s revenue in singleitem singleunit Bayesian posted price auctions, focusing on two different settings: public signaling, where the signals are publicly visible to all buyers, and private signaling, in which the seller can send a different signal to each buyer through private communication channels. As a first negative result, we prove that, in both settings, the problem does not admit an FPTAS unless , even for basic instances with a single buyer. Then, we provide tight positive results by designing a PTAS for each setting. In order to do so, we first introduce a unifying framework encompassing both public and private signaling. Its core result is a decomposition lemma that allows us to focus on a finite set of buyers’ posterior beliefs over states of nature—called uniform posteriors—, rather than reasoning about signaling schemes with a (potentially) infinite number of signals. Compared to previous works on signaling, our framework has to deal with some additional challenges. The main one is that, in our model, the seller (sender) is not only required to choose how to send signals, but they also have to take some actions in the form of price proposals. This requires significant extensions to standard approaches based on decomposition lemmas (Cheng et al., 2015; Xu, 2020; Castiglioni and Gatti, 2021)
. The framework forms the basis on which we design our PTASs. In the public setting, it establishes a connection between signaling schemes and probability distributions over
uniform posteriors. This allows us to formulate the seller’s revenuemaximizing problem as an LP of polynomial size, whose objective coefficients are notreadily available. However, they can be approximately computed in polynomial time by an algorithm for finding approximatelyoptimal prices in (nonBayesian) posted price auctions, which may also be of independent interest.Solving the LP with approximate coefficients then gives the desired PTAS. As for the private setting, our framework provides a connection between marginal signaling schemes of each buyer and probability distributions over
uniform posteriors, which, to the best of our knowledge, is the first of its kind, since previous works are limited to public settings (Cheng et al., 2015; Castiglioni et al., 2020b).^{1}^{1}1A notable exception is (Castiglioni and Gatti, 2021), which studies a specific case in between private and public signaling schemes. Such connection allows us to formulate an LP correlating marginal signaling schemes together and with price proposals. Although the LP has an exponential number of variables, we show that it can still be approximately solved in polynomial time by means of the ellipsoid method. This requires the implementation of a problemspecific approximate separation oracle that can be implemented in polynomial time by means of a dynamic programming algorithm.Related Works.
The computational study of Bayesian persuasion has received terrific attention (Vasserman et al., 2015; Castiglioni et al., 2020a; Rabinovich et al., 2015; Candogan, 2019; Castiglioni et al., 2022, 2021). The works most related to ours are those addressing secondprice auctions. Emek et al. (2014) provide an LP to compute an optimal public signaling scheme in the knownvaluation setting, and they show that the problem is hard in the Bayesian setting. Cheng et al. (2015) provide a PTAS for this latter case. Bacchiocchi et al. (2022) extend the framework to study ad auctions with Vickrey–Clarke–Groves payments. Finally, Badanidiyuru et al. (2018) focus on the design of algorithms whose running time is independent from the number of states of nature. They initiate the study of private signaling, showing that, in secondprice auctions, it may introduce nontrivial equilibrium selection issues.
2 Preliminaries
2.1 Bayesian Posted Price Auctions and Signaling
In a posted price auction, the seller tries to sell an item to a finite set of buyers arriving sequentially according to a fixed ordering. W.l.o.g., we let buyer be the th buyer according to such ordering. The seller chooses a price proposal for each buyer . Then, each buyer in turn has to decide whether to buy the item for the proposed price or not. Buyer buys only if their item valuation is at least the proposed price .^{2}^{2}2As customary in the literature, we assume that buyers always buy when they are offered a price that is equal to their valuation. In that case, the auction ends and the seller gets revenue for selling the item, otherwise the auction continues with the next buyer.
We study Bayesian posted price auctions, characterized by a finite set of states of nature, namely . Each buyer
has a valuation vector
, with representing buyer ’s valuation when the state is . Each valuation is independently drawn from a probability distribution supported on . For the ease of presentation, we let be the matrix of buyers’ valuations, whose entries are for all and .^{3}^{3}3Sometimes, we also write to denote the th row of matrix , which is the valuation of buyer . Moreover, by letting be the collection of all distributions of buyers’ valuations, we write to denote that is built by drawing each independently from .We model signaling with the Bayesian persuasion framework by Kamenica and Gentzkow (2011). We consider the case in which the seller—having knowledge of the state of nature—acts as a sender by issuing signals to the buyers (the receivers), so as to partially disclose information about the state and increase revenue. As customary in the literature, we assume that the state is drawn from a common prior distribution , explicitly known to both the seller and the buyers.^{4}^{4}4In this work, given a finite set , we denote with the ()dimensional simplex defined over the elements of . We denote by the probability of state . The seller commits to a signaling scheme , which is a randomized mapping from states of nature to signals for the receivers. Letting be the set of signals for buyer , a signaling scheme is a function , where . An element —called signal profile—is a tuple specifying a signal for each buyer. We use to refer to the th component of any (i.e., the signal for buyer ), so that . We let be the probability of drawing signal profile when the state is . Furthermore, we let be the marginal signaling scheme of buyer , with being the marginalization of with respect to buyer ’s signals. As for general signaling schemes, denotes the probability of drawing signal when the state is .
Price proposals may depend on the signals being sent to the buyers. Formally, the seller commits to a price function , with being the price vector when the signal profile is . We assume that prices proposed to buyer only depend on the signals sent to them, and not on the signals sent to other buyers. Thus, w.l.o.g., we can work with functions defining prices for each buyer independently, with denoting the th component of for all and .^{5}^{5}5Let us remark that our assumption on the seller’s price function ensures that a buyer does not get additional information about the state of nature by observing the proposed price, since the latter only depends on the signal which is revealed to them anyway.
The interaction involving the seller and the buyers goes on as follows (Figure 1): (i) the seller commits to a signaling scheme and a price function , and the buyers observe such commitments; (ii) the seller observes the state of nature ; (iii) the seller draws a signal profile ; and (iv) the buyers arrive sequentially, with each buyer observing their signal and being proposed price . Then, each buyer rationally updates their prior belief over states according to Bayes rule, and buys the item only if their expected valuation for the item is greater than or equal to the offered price. The interaction terminates whenever a buyer decides to buy the item or there are no more buyers arriving. The following paragraph formally defines the elements involved in step (iv).
Buyers’ Posteriors.
In step (iv), a buyer receiving a signal infers a posterior belief over states (also called posterior), which we denote by , with
being the posterior probability of state
. Formally,(1) 
Thus, after receiving signal , buyer ’s expected valuation for the item is , and the buyer buys it only if such value is at least as large as the price . In the following, given a signal profile , we denote by a tuple defining all buyers’ posteriors resulting from observing signals in ; formally, .
Distributions on Posteriors.
In singlereceiver Bayesian persuasion models, it is oftentimes useful to represent signaling schemes as convex combinations of the posteriors they can induce. In our setting, a marginal signaling scheme of buyer induces a probability distribution over posteriors in , with denoting the probability of posterior . Formally, it holds that
Intuitively, denotes the probability that buyer has posterior . Indeed, it is possible to directly reason about distributions rather than marginal signaling schemes, provided that such distributions are consistent with the prior. Formally, by letting be the support of , it must be required that
(2) 
2.2 Computational Problems
We focus on the problem of computing a signaling scheme and a price function that maximize the seller’s expected revenue, considering both public and private signaling settings.^{6}^{6}6Formally, a signaling scheme is public if: (i) for all ; and (ii) for every , only for signal profiles such that for . Since, given a signal profile , under a public signaling scheme all the buyers always share the same posterior (i.e., for all ), we overload notation and sometimes use to denote the unique posterior appearing in . Similarly, in the public setting, given a posterior we sometimes write in place of a tuple of copies of .
We denote by the expected revenue of the seller when the distributions of buyers’ valuations are given by , the proposed prices are defined by the vector , and the buyers’ posteriors are those specified by the tuple containing a posterior for each buyer . Then, the seller’s expected revenue is:
In the following, we denote by the value of the seller’s expected revenue for a revenuemaximizing pair.
In this work, we assume that algorithms have access to a blackbox oracle to sample buyers’ valuations according to the probability distributions specified by (rather than actually knowing such distributions). Thus, we look for algorithms that output pairs such that
where is an additive error. Notice that the expectation above is with respect to the randomness of the algorithm, which originates from using the blackbox sampling oracle.
3 Hardness of Signaling with a Single Buyer
We start with a negative result: there is no FPTAS for the problem of computing a revenuemaximizing pair unless , in both public and private signaling settings. Our result holds even in the basic case with only one buyer, where public and private signaling are equivalent. Notice that, in the reduction that we use to prove our result, we assume that the support of the distribution of valuations of the (single) buyer is finite and that such distribution is perfectly known to the seller. This represents an even simpler setting than that in which the seller has only access to a blackbox oracle returning samples drawn from the buyer’s distribution of valuations. The result formally reads as follows:
Theorem 1.
There is no additive FPTAS for the problem of computing a revenuemaximizing pair unless , even when there is a single buyer.
4 Unifying Public and Private Signaling
In this section, we introduce a general mathematical framework related to buyers’ posteriors and distributions over them, proving some results that will be crucial in the rest of this work, both in public and private signaling scenarios.
One of the main difficulties in computing senderoptimal signaling schemes is that they might need a (potentially) infinite number of signals, resulting in infinitelymany receiver’s posteriors. The trick commonly used to circumvent this issue in settings with a finite number of valuations is to use direct signals, which explicitly specify action recommendations for each receiver’s valuation Castiglioni et al. (2020c, 2021). However, in our auction setting, this solution is not viable, since a direct signal for a buyer should represent a recommendation for every possible , and these are infinitely many. An alternative technique, which can be employed in our setting, is to restrict the number of possible posteriors.
Our core idea is to focus on a small set of posteriors, which are those encoded as particular uniform probability distributions, as formally stated in the following definition.^{7}^{7}7In all the definitions and results of this section (Section 4), we denote by a generic posterior common to all the buyers and with a probability distribution over (i.e, over posteriors).
Definition 1 (uniform posterior).
A posterior is uniform if it can be obtained by averaging the elements of a multiset defined by canonical basis vectors of .
We denote the set of all uniform posteriors as . Notice that the set has size .
The existence of an approximatelyoptimal signaling scheme that only uses uniform posteriors is usually proved by means of socalled decomposition lemmas (see (Cheng et al., 2015; Xu, 2020; Castiglioni and Gatti, 2021)). The goal of these lemmas is to show that, given some signaling scheme encoded as a distribution over posteriors, it is possible to obtain a new signaling scheme whose corresponding distribution is supported only on uniform posteriors, and such that the sender’s utility only decreases by a small amount. At the same time, these lemmas must also ensure that the distribution over posteriors corresponding to the new signaling scheme is still consistent (according to Equation (2)).
The main result of our framework (Theorem 2) is a decomposition lemma that is suitable for our setting. Before stating the result, we need to introduce some preliminary definitions.
Definition 2 (decreasing distribution).
Let . A probability distribution over is decreasing around a given posterior if the following condition holds for every matrix of buyers’ valuations:
Intuitively, a probability distribution as in Definition 2 can be interpreted as a perturbation of the given posterior such that, with high probability, buyers’ expected valuations in are at most less than those in posterior .^{8}^{8}8Definition 2 is similar to analogous ones in the literature (Xu, 2020; Castiglioni and Gatti, 2021), where the distance is usually measured in both directions, as . We look only at the direction of decreasing values, since in a our setting, if a buyer’s valuation increases, then the seller’s revenue also increases.
The second definition we need is about functions mapping vectors in —defining a valuation for each buyer—to seller’s revenues. For instance, one such function could be the seller’s revenue given price vector . In particular, we define the stability of a function compared to another function . Intuitively, is stable compared to if the value of , in expectation over buyers’ valuations and posteriors drawn from a probability distribution that is decreasing around , is “close” to the the value of given , in expectation over buyers’ valuations.^{9}^{9}9The notion of compared stability has been already used (Cheng et al., 2015; Castiglioni and Gatti, 2021). However, previous works consider the case in which is a relaxation of . Instead, our definition is conceptually different, as and represent two different functions corresponding to different price vectors of the seller. Formally:
Definition 3 (stability).
Let . Given a posterior , some distributions , and two functions , is stable compared to for if, for every probability distribution over that is decreasing around , it holds:
Now, we are ready to state our main result. We show that, for any buyer’s posterior , if a function is stable compare to , then there exists a suitable probability distribution over uniform posteriors such that the expected value of given such distribution is “close” to that of given .
Theorem 2.
Let , and set . Given a posterior , some distributions , and two functions , if is stable compared to for , then there exists such that, for every , and
(3) 
The crucial feature of Theorem 2 is that Equation (3) holds for every state. This is fundamental for proving our results in the private signaling scenario. On the other hand, with public signaling, we will make use of the following (weaker) corollary, obtained by summing Equation (3) over all .
Corollary 1.
Let , and set . Given a posterior , some distributions , and two functions , if is stable compared to for , then there exists such that, for every , and
(4) 
5 Warming Up: NonBayesian Auctions
In this section, we focus on nonBayesian posted price auctions, proving some results that will be useful in the rest of the paper.^{10}^{10}10When we study nonBayesian posted price auctions, we stick to our notation, with the following differences: valuations are scalars rather than vectors, namely ; distributions are supported on rather than ; the matrix is indeed a column vector whose components are buyers’ valuations; and the price function is replaced by a single price vector , with its th component being the price for buyer . Moreover, we continue to use the notation Rev to denote seller’s revenues, dropping the dependence on the tuple of posteriors. Thus, in a nonBayesian auction in which the distributions of buyers’ valuations are , the notation simply denotes the seller’s expected revenue by selecting a price vector . In particular, we study what happens to the seller’s expected revenue when buyers’ valuations are “slightly decreased”, proving that the revenue also decreases, but only by a small amount. This result will be crucial when dealing with public signaling, and it also allows to design a polytime algorithm for finding approximatelyoptimal price vectors in nonBayesian auctions, as we show at the end of this section.
In the following, we extensively use distributions of buyers’ valuations as specified in the definition below.
Definition 4.
Given , we denote by and two collections of distributions of buyers’ valuations such that, for every price vector ,
Intuitively, valuations drawn from are “slightly decreased” with respect to those drawn from , since the probability with which any buyer buys the item at the (reduced) price when their valuation is drawn from is at least as large as the probability of buying at price when their valuation is drawn from .^{11}^{11}11In this work, given , we let . We extend the operator to vectors by applying it componentwise.
Our main contribution in this section (Lemma 2) is to show that . By letting be any revenuemaximizing price vector under distributions , one may naïvely think that, since under distributions and price vector each buyer would buy the item at least with the same probability as with distributions and price vector , while paying a price that is only less, then , proving the result. However, this line of reasoning does not work, as shown by Example 1 in the Extended Version. The crucial feature of Example 1 is that there exists a in which one buyer is offered a price that is too low, and, thus, the seller prefers not to sell the item to them, but rather to a following buyer. This prevents a direct application of the line of reasoning outlined above, as it shows that incrementing the probability with which a buyer buys is not always beneficial. One could circumvent this issue by considering a such that the seller is never upset if some buyer buys. In other words, it must be such that each buyer is proposed a price that is at least as large as the seller’s expected revenue in the posted price auction restricted to the following buyers. Next, we show that there always exists a with such desirable property.
Letting be the seller’s revenue for price vector and distributions in the auction restricted to buyers , we prove the following:
Lemma 1.
For any , there exists a revenuemaximizing price vector such that for every buyer .
The proof of Lemma 2 builds upon the existence of a revenuemaximizing price vector as in Lemma 1 and the fact that, under distributions , the probability with which each buyer buys the item given price vector is greater than that with which they would buy given . Since the seller’s expected revenue is larger when a buyer buys compared to when they do not buy (as ), the seller’s expected revenue decreases by at most .
Lemma 2.
Given , let and satisfying the conditions of Definition 4. Then, .
Lemma 2 will be useful to prove Lemma 3 and to show the compared stability of a suitablydefined function that is used to design a PTAS in the public signaling scenario.
Finding ApproximatelyOptimal Prices.
Algorithm 1 computes (in polynomial time) an approximatelyoptimal price vector for any nonBayesian posted price auction. It samples matrices of buyers’ valuations, each one drawn according to the distributions . Then, it finds an optimal price vector in the discretized set , assuming that buyers’ valuations are drawn according to the empirical distribution resulting from the sampled matrices.^{12}^{12}12In this work, for a discretization step , we let be the set of prices multiples of , while . This last step can be done by backward induction, as it is well known in the literature (see, e.g., (Xiao et al., 2020)). The following Lemma 3 establishes the correctness of Algorithm 1
, also providing a bound on its running time. The key ideas of its proof are: (i) the sampling procedure constructs a good estimation of the actual distributions of buyers’ valuations; and (ii) even if the algorithm only considers discretized prices, the components of the computed price vector are at most
less than those of an optimal (unconstrained) price vector. As shown in the proof, this is strictly related to reducing buyer’s valuations by . Thus, it follows by Lemma 2 that the seller’s expected revenue is at most less than the optimal one.Lemma 3.
For any and , there exist and such that, with probability at least , Algorithm 1 returns satisfying and in time .
6 Public Signaling
In the following, we design a PTAS for computing a revenuemaximizing pair in the public signaling setting. Notice that this positive result is tight by Theorem 1.
As a first intermediate result, we prove the compared stability of suitablydefined functions, which are intimately related to the seller’s revenue. In particular, for every price vector , we conveniently let be a function that takes a vector of buyers’ valuations and outputs the seller’s expected revenue achieved by selecting when the buyers’ valuations are those specified as input. The following Lemma 4 shows that, given some distributions of buyers’ valuations and a posterior , there always exists a price vector such that is stable compared with for every other . This result crucially allows us to decompose any posterior by means of the decomposition lemma in Corollary 1, while guaranteeing a small loss in terms of seller’s expected revenue.
Lemma 4.
Given , a posterior , and some distributions of buyers’ valuations , there exists such that, for every other , the function is stable compared with for .
Our PTAS leverages the fact that public signaling schemes can be represented as probability distributions over buyers’ posteriors (recall that, in the public signaling setting, all the buyers share the same posterior, as they all observe the same signal). In particular, the algorithm returns a pair , where is a probability distribution over satisfying consistency constraints (see Equation (2)), while is a function mapping each posterior to a price vector. In singlereceiver settings, it is well known (see Subsection 2.1) that using distributions over posteriors rather than signaling schemes is without loss of generality. The following lemma shows that the same holds in our case, i.e., given a pair , it is always possible to obtain a pair providing the seller with the same expected revenue.
Lemma 5.
Given a pair , where is a probability distribution over with for all and , there is a pair s.t.
Next, we show that, in order to find an approximatelyoptimal pair , we can restrict the attention to uniform posteriors (with suitably defined). First, we introduce the following LP that computes an optimal probability distribution restricted over uniform posteriors.
(5a)  
(5b) 
The following Lemma 6 shows the optimal value of LP 5 is “close” to . Its proof is based on the following core idea. Given the signaling scheme in a revenuemaximizing pair , letting be the distribution over induced by , we can decompose each posterior in the support of according to Corollary 1. Then, the obtained distributions over uniform posteriors are consistent according to Equation (2), and, thus, they satisfy Constraints (5b). Moreover, since such distributions are also decreasing around the decomposed posteriors, by Lemma 4 each time a posterior is decomposed there exists a price vector resulting in a small revenue loss. These observations allow us to conclude that the seller’s expected revenue provided by an optimal solution to LP 5 is within some small additive loss of .
Lemma 6.
Given and letting , an optimal solution to LP 5 has value at least .
Finally, we are ready to provide our PTAS. Its main idea is to solve LP 5 (of polynomial size) for the value of in Lemma 6. This results in a small revenue loss. The last part missing for the algorithm is computing the terms appearing in the objective of LP 5, i.e., a revenuemaximizing price vector (together with its revenue) for every uniform posterior. In order to do so, we can use Algorithm 1 (see also Lemma 3), which allows us to obtain in polynomial time good approximations of such price vectors, with high probability.
Theorem 3.
There exists an additive PTAS for computing a revenuemaximizing pair with public signaling.
7 Private Signaling
With private signaling, computing a pair amounts to specifying a pair for each buyer —composed by a marginal signaling scheme and a price function for buyer —, and, then, correlating the so as to obtain a (nonmarginal) signaling scheme . We leverage this fact to design our PTAS.
In Subsection 7.1, we first show that it is possible to restrict the set of marginal signaling schemes of a given buyer to those encoded as distributions over uniform posteriors, as we did with public signaling. Then, we provide an LP formulation for computing an approximatelyoptimal pair, dealing with the challenge of correlating marginal signaling schemes in a nontrivial way. Finally, in Subsection 7.2, we show how to compute a solution to the LP in polynomial time, which requires the application of the ellipsoid method in a nontrivial way, due to the features of the formulation.
7.1 LP for Approximate Signaling Schemes
Before providing the LP, we show that restricting marginal signaling schemes to uniform posteriors results in a buyer’s behavior which is similar to the one with arbitrary posteriors. This amounts to showing that suitablydefined functions related to the probability of buying are comparatively stable.
For and , let be a function that takes as input a vector of buyers’ valuations and outputs if and only if (otherwise it outputs ).
Lemma 7.
Given and some distributions , for every buyer , posterior , and price , the function is stable compared with for .
The following remark will be crucial for proving Lemma 9. It shows that, if for every we decompose buyer ’s posterior by means of a distribution over uniform posteriors decreasing around , then the probability with which buyer buys only decreases by a small amount.^{13}^{13}13In this section, for the ease of presentation, we abuse notation and use to denote the (all equal) sets of uniform posteriors (Definition 1), one per buyer , while is the set of tuples specifying a for each .
Remark 1.
Next, we show that an approximatelyoptimal pair can be found by solving LP 6 instantiated with suitablydefined and . LP 6 employs:

Variables (for and ), which encode the distributions over posteriors representing the marginal signaling schemes of the buyers.

Variables (for , , and ), with encoding the probability that the seller offers price to buyer and buyer ’s posterior is .

Variables (for , , and ), with encoding the probability that the state is , the buyers’ posteriors are those specified by , and the prices that the seller offers to the buyers are those given by .
(6a)  
(6b)  
(6c)  
(6d) 
Variables represent marginal signaling schemes, allowing for multiple signals inducing the same posterior. This is needed since signals may correspond to different price proposals.^{14}^{14}14Notice that, in a classical setting in which the sender does not have to propose a price (or, in general, select some action after sending signals), there always exists a signaling scheme with no pair of signals inducing the same posterior. Indeed, two signals that induce the same posterior can always be joined into a single signal. This is not the case in our setting, where we can only join signals that induce the same posterior and correspond to the same price. One may think of marginal signaling schemes in LP 6 as if they were using signals defined as pairs , with the convention that . Variables and Constraints (6b) ensure that marginal signaling schemes are correctly correlated together, by directly working in the domain of the distributions over posteriors.
To show that an optimal solution to LP 6 provides an approximatelyoptimal pair, we need the following two lemmas. Lemma 8 proves that, given a feasible solution to LP 6, we can recover a pair providing the seller with an expected revenue equal to the value of the LP solution. Lemma 9 shows that the optimal value of LP 6 is “close” to . These two lemmas imply that an approximatelyoptimal pair can be computed by solving LP 6.
Lemma 8.
Given a feasible solution to LP 6, it is possible to recover a pair that provides the seller with an expected revenue equal to the value of the solution.
Lemma 9.
For every , there exist such that LP 6 has optimal value at least .
7.2 Ptas
We provide an algorithm that approximately solves LP 6 in polynomial time, which completes our PTAS for computing a revenuemaximizing pair in the private setting. The core idea of our algorithm is to apply the ellipsoid method on the dual of LP 6.^{15}^{15}15To be precise, we apply the ellipsoid method to the dual of a relaxed version of LP 6, since we need an overconstrained dual. More details on these technicalities are in the Extended Version. In particular, our implementation of the ellipsoid algorithm uses an approximate separation oracle that needs to solve the following optimization problem.
Definition 5 (MaxLinrev).
Given some distributions of buyers’ valuations such that each has finite support and a vector , solve
As a first step, we provide an FPTAS for MAXLINREV using a dynamic programming approach. This will be the main building block of our approximate separation oracle.^{16}^{16}16Notice that, since MAXLINREV takes as input distributions with a finite support, we can safely assume that such distributions can be explicitly represented in memory. In our PTAS, the inputs to the dynamic programming algorithm are obtained by building empirical distributions through samples from the actual distributions of buyers’ valuations, thus ensuring finiteness of the supports.
The FPTAS works as follows. Given an error tolerance , it first defines a step size , with , and builds a set of possible discretized values for the linear term appearing in the MAXLINREV objective. Then, for every buyer (in reversed order) and value , the algorithm computes , which is an approximation of the largest seller’s revenue provided by a pair when considering buyers only, and restricted to pairs such that the inequality is satisfied. By letting , the value can be defined by the following recursive formula:^{17}^{17}17Notice that, given a pair with and , it is possible to compute in polynomial time the probability with which a buyer buys the item.
Finally, the algorithm returns . Thus:
Lemma 10.
For any , there exists a dynamic programming algorithm that provides a approximation (in the additive sense) to MAXLINREV. Moreover, the algorithm runs in time polynomial in the size of the input and .
Now, we are ready to prove the main result of this section.
Theorem 4.
There exists an additive PTAS for computing a revenuemaximizing pair with private signaling.
Acknowledgments
This work has been partially supported by the Italian MIUR PRIN 2017 Project ALGADIMAR “Algorithms, Games, and Digital Market”.
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Appendix A Additional Discussion on Lemma 2
Our main contribution in Section 5 (Lemma 2) is to show that: . By letting be a revenuemaximizing price vector for the seller under distributions , one may naïvely think of proving the result by simply showing that, given the price vector with and distributions , each buyer would buy the item at least with the same probability as for price vector and distributions , but paying a price that is only less. This would imply that , proving the desired result. However, this line of reasoning does not work, since, as shown by the following example, it has a major flaw.
Example 1.
Consider a posted price auction with two buyers. In the first case (distributions ), buyer 1 has valuation and buyer 2 has valuation . In such setting, an optimal price vector is such that and , so that the revenue of the seller, namely , is . In the second case (distributions ), buyer 1 has valuations and buyer 2 has valuation . Thus, the revenue of the seller for the price vector (with and ), namely , is , since buyer 1 will buy the item.
The crucial feature of the setting described in Example 1 is that there is an optimal price vector in which one buyer (buyer 1) is offered a price that is too low, and, thus, the seller prefers not to sell the item to them, but rather to another buyer (buyer 2). This prevents a direct application of the line of reasoning outlined above. However, one could circumvent this issue by considering an optimal price vector such that the seller is never upset if some buyer buys. In other words, prices must be such that each buyer is proposed a price that is at least as large as the seller’s expected revenue in the posted price auction restricted to buyers following them. In Example 1, the optimal price vector such that would be fine.
Appendix B Proofs omitted from Section 3
See 1
Proof.
We employ a reduction from an hard problem originally introduced by Khot and Saket (2012), which we formally state in the following. For any positive integer , integer such that , and arbitrarily small constant , the problem reads as follows. Given an undirected graph , distinguish between:

Case 1. There exists a colorable induced subgraph of containing a fraction of all vertices, where each color class contains a fraction of all vertices.^{18}^{18}18A colorable induced subgraph is identified by a subset of vertices such that it is possible to assign one among different colors to each vertex, in such a way that there are no two adjacent vertices having the same color. Given some color, its associated color class is the subset of all vertices in the subgraph having that color.

Case 2. Every independent set of contains less than a fraction of all vertices.^{19}^{19}19An independent set of is a subset of vertices such that there are no two adjacent vertices.
We reduce from such problem for , and . Our reduction works as follows:

Completeness. If Case 1 holds, then there exists a signaling scheme, price function pair that provides the seller with an expected revenue at least as large as some threshold (see Equation (7) below for its definition).

Soundness. If Case 2 holds, then the seller’s expected revenue for any signaling scheme, price function pair is smaller than with , where denotes the number or vertices of the graph .
This shows that it is hard to approximate the optimal seller’s expected revenue up to within an additive error . Thus, since depends polynomially on the size of the problem instance, this also shows that there is no additive FPTAS for the problem of computing a revenuemaximizing pair, unless .
Construction
Given an undirected graph , with vertices , we build a singlebuyer Bayesian posted price auction as follows.^{20}^{20}20In a singlebuyer setting, we always omit the subscript from symbols, as it is clear that they refer to the unique buyer. Moreover, with an overload of notation, we use buyer’s signals as if they were signal profiles. There is one state of nature for each vertex , and the prior belief over states is such that for all . There is a finite set of possible buyer’s valuations. For every vertex , there is a valuation vector such that:

;

for all ; and

for all .
Each valuation has probability of occurring according to the distribution . Moreover, there is an additional valuation vector such that for all , having probability .
Completeness
Assume that a colored induced subgraph of is given, and that it contains a fraction of vertices, while each color class is made up of a fraction of all vertices. We let