Bilateral Trade with Correlated Values

We study the bilateral trade problem where a seller owns a single indivisible item, and a potential buyer seeks to purchase it. Previous mechanisms for this problem only considered the case where the values of the buyer and the seller are drawn from independent distributions. In this paper, we study bilateral trade mechanisms when the values are drawn from a joint distribution. We prove that the buyer-offering mechanism guarantees an approximation ratio of $\frac e {e-1} \approx 1.582$ to the social welfare even if the values are drawn from a joint distribution. The buyer-offering mechanism is Bayesian incentive compatible, but the seller has a dominant strategy. We prove the buyer-offering mechanism is optimal in the sense that no Bayesian mechanism where one of the players has a dominant strategy can obtain an approximation ratio better than $\frac e {e-1}$. We also show that no mechanism in which both sides have a dominant strategy can provide any constant approximation to the social welfare when the values are drawn from a joint distribution. Finally, we prove some impossibility results on the power of general Bayesian incentive compatible mechanisms. In particular, we show that no deterministic Bayesian incentive-compatible mechanism can provide an approximation ratio better than $1+\frac {\ln 2} 2\approx 1.346$.


Introduction
This paper focuses on the bilateral trade problem where a seller owns a single indivisible item, and a potential buyer seeks to purchase it.The seller has a value of s ≥ 0 associated with retaining the item (and 0 otherwise), whereas the buyer's value for obtaining it is b ≥ 0 (if the buyer does not receive the item, then the buyer's value is 0).The values b and s are private, but the probability distributions from which they were derived are known.
The two most common objectives in bilateral trade scenarios are to maximize the social welfare and to maximize the gains from trade.The former goal is aimed at maximizing the total value generated by the transaction (that is, the social welfare is b in case of trade and s otherwise), while the latter is focused on increasing the difference between the buyer's and the seller's surplus (i.e., the gains from trade is b − s in case of trade, and 0 otherwise).Our interest in this paper is in incentive-compatible mechanisms that are strongly budget balanced.That is, the buyer's payment is fully transferred to the seller.See Section 2 for a precise statement of the problem.
The problem was extensively studied, and here, we only mention some of the papers closest to our research.Myerson and Satterthwaite [MS83] prove that under very mild conditions, no Bayesian mechanism can exactly maximize the gains from trade or, equivalently, the social welfare.Blumrosen and Dobzinski [BD14] show that a fixed-price mechanism guarantees at least half of the optimal social welfare (equivalently, provides a 2-approximation) for every distribution.This approximation ratio was later improved to 1.99 [CBdKLT16], then to e e−1 [BD21], then to e e−1 − 0.0001 [KPV22], and then to an almost optimal ratio of approximately 1.38 [CW23,LRW23].
There has also been much work regarding approximating the gains from trade.McAfee [McA08] shows that for some distributions of the buyer and seller's values, there exists a fixed price mechanism that recovers half of the optimal gains-from-trade.However, for every fixed fraction, there are distributions for which any fixed price guarantees less than this fraction of the optimal gains-from-trade [BD14].To overcome this, Blumrosen and Mizrahi [BM16] propose the seller-offering mechanism, in which the seller makes the buyer the profit-maximizing take-it-or-leave-it offer, given his value s.In the seller-offering mechanism, only the buyer has a dominant strategy.Still, they show that if the buyer's value is drawn from a distribution with a monotone hazard rate, then the seller-offering mechanism is Bayesian incentive compatible and provides an e approximation to the optimal gains-from-trade.Brustle et al. [BCWZ17] show that the better of the seller offering mechanism and the buyer offering mechanism (in which the buyer makes the seller a profit-maximizing take-it-or-leave-it-offer) recovers at least half of the gains from trade of every incentive-compatible mechanism.In a breakthrough result, Deng et al. [DMSW22] show that the better of the offering mechanisms provides 8.23 approximation to the optimal gains-from-trade.This constant was later improved to 3.15 by Fei [Fei22].We also know that the better of the offering mechanisms sometimes recovers strictly less than half of the optimal gains from trade [BDK21].
Despite this extensive work, existing research on bilateral trade, including the works cited above, largely assumes that the values of the seller and the buyer are drawn independently 1 .This paper investigates a more realistic and technically challenging scenario in which the values are derived from a joint probability distribution.Remarkably, we demonstrate that despite the correlation, the buyer-offering mechanism approximates the social welfare within a constant factor.Furthermore, this factor is quite close to the best approximation ratio possible for independent distributions.Theorem: For every joint distribution of the seller and buyer's values, there is a mechanism that provides an e e−1 -approximation to the optimal social welfare 2 .Interestingly, whereas the power of the buyer-offering mechanism is equal to the power of the seller-offering mechanism for gains-from-trade approximation, the buyer-offering mechanism is much more powerful than the seller-offering mechanism in our setting: it is not hard to see that the seller-offering mechanism does not guarantee any constant fraction of the optimal social welfare 3 .
The buyer-offering mechanism is natural and simple.It is also appealing from a theoretical point of view: it is not only Bayesian incentive compatible, but in fact, one player (the offered player) has a dominant strategy as it can only accept or reject a take-it-or-leave-it offer.We call Bayesian incentive compatible mechanisms in which one side has a dominant strategy one-sided dominant strategy mechanisms.We prove that the buyer-offering mechanism is optimal among all one-sided mechanisms: Theorem: There exists a joint distribution of the seller and buyer's values such that the approximation ratio of every one-sided dominant strategy mechanism is no better than e e−1 .This theorem demonstrates, in particular, that even taking, e.g., the better of the seller-offering and the buyeroffering mechanism, or the better of the buyer-offering mechanism and some fixed price mechanism, does not improve the approximation guarantee of the buyer-offering mechanism.
Moreover, we show that mechanisms in which both sides have a dominant strategy cannot guarantee any fixed fraction of the optimal social welfare 4 .
Theorem: For every constant c > 1, there exists a joint distribution of the seller and buyer's values such that the approximation ratio of every dominant strategy incentive compatible mechanism is at least c.
Unlike the impossibility of one-sided mechanisms (which requires careful analysis and subtle construction), the proof that dominant-strategy mechanisms have no power is technically simpler.We then continue analyzing the power of Bayesian incentive-compatible mechanisms, now proving limits on their power: Theorem: There exists a joint distribution of the seller and buyer's values such that no deterministic Bayesian incentive-compatible mechanism provides a better than 1 + ln 2 2 ≈ 1.346 approximation to the optimal social welfare.
Previously, Blumrosen and Mizrahi [BM16] proved a bound of 1.07 on the approximation ratio of Bayesian mechanisms for independent distributions.Their proof relies on characterizing the second-best mechanism of [MS83].In our setting, proving impossibilities with this approach is less promising: not only is the Myerson-Satterthwaite characterization often hard to compute for independent distributions, but it also does not apply to joint distributions.Thus, we present a new technique for proving impossibilities for Bayesian incentivecompatible mechanisms.Our approach is based on introducing and analyzing a certain family of "L-shaped" distributions, for which the second-best mechanism has a nicer structure.A disadvantage of our approach is that our results only apply to deterministic Bayesian mechanisms that are ex-post individually rational, whereas the results of Blumrosen and Mizrahi apply to randomized mechanisms that are interim individually rational ones.We note that all major mechanisms considered in the literature (e.g., fixed price mechanisms, buyer and seller offering mechanisms) are deterministic and ex-post individually rational 5 .Nevertheless, our technique is robust enough that, as a by-product, we improve the state-of-the-art impossibilities for both the social welfare and gains-from-trade even with independent distributions: Theorem: • There exist independent distributions of the seller and buyer's values such that no deterministic Bayesian incentive-compatible mechanism provides an approximation ratio better than 2-approximation to the optimal gains from trade.
• There exist independent distributions of the seller and buyer's values such that no deterministic Bayesian incentive-compatible mechanism provides an approximation ratio better than 1.113-fraction to the optimal social welfare.
3 Consider a seller with a fixed value s = 0 and a buyer that its value is distributed by an equal revenue distribution in [1, k].The optimal welfare is ln k whereas the welfare of the seller offering mechanism is only 1.
4 A simple observation is that mechanisms in which both sides have a dominant strategy are fixed price mechanisms [BD14]. 5Our impossibilities also apply to mechanisms which are a probability distribution over deterministic ex-post individually rational mechanisms, like the random offerer mechanism [DMSW22].To see this, consider such a mechanism A. A simple averaging argument shows that in the support of A there must be a (deterministic and ex-post individually rational) mechanism A ′ that its approximation ratio is at least the approximation ratio of A. Our impossibilities directly apply to A ′ , and the approximation ratio of A is no better than the approximation ratio of A ′ .

Open Questions and Future Directions
In this paper, we analyzed the power of incentive-compatible mechanisms for bilateral trade.We proved that the buyer-offering mechanism provides an approximation ratio of e e−1 ≈ 1.582 even if the values are drawn from joint distributions.We proved that this ratio is optimal for one-sided mechanisms and that dominant strategy mechanisms cannot guarantee any fixed fraction of the welfare at all.However, there is a gap between this ratio and our impossibility result of 1 + ln 2 2 ≈ 1.346.We leave closing this gap as an open question.It will also be interesting to understand the power of Bayesian incentive compatible mechanisms for independent distributions and determine whether they are more powerful than deterministic mechanisms [LRW23,CW23].
Another question that remains open is to determine whether Bayesian incentive-compatible mechanisms can give a constant approximation to the gains-from-trade even when the values are drawn from a joint distribution.
Lastly, all our impossibility results for Bayesian incentive-compatible mechanisms hold only for deterministic Bayesian incentive-compatible mechanisms.In Section B, we show that there exist distributions for which randomized Bayesian incentive-compatible mechanisms outperform deterministic Bayesian incentivecompatible mechanisms.An important future direction is understanding the power of randomized Bayesian incentive-compatible mechanisms in all models discussed in the paper.
Structure of the Paper.In Section 2 we give the necessary preliminaries.In Section 3, we prove that the buyer-offering mechanism provides an e e−1 -approximation, even for correlated distributions.Section 4 shows that e e−1 is the best ratio achievable by one-sided dominant strategy mechanism.In Section 5, we prove several impossibilities for Bayesian incentive-compatible mechanisms.In Appendix A, we show that no two-sided dominant strategy incentive compatible mechanism provides a bounded approximation ratio.

The Setting
In the bilateral trade problem, we have two agents: the seller and the buyer.The seller owns an indivisible item and his value for it is s.The buyer's value for the item is b.The values b and s are drawn from a joint distribution F .
A (direct) deterministic mechanism M for the bilateral trade problem consists of two functions M = (x, p).For every tuple of seller and buyer values (s, b) in the support of F , x(s, b) = 1 if a trade occurs and x(s, b) = 0 otherwise.If there is a trade, p(s, b) specifies the price that the buyer pays for the item and the payment that the seller gets for it.We require that b ≥ p(s, b) ≥ s.These restrictions on the payment are called ex-post individual rationality.
The optimal welfare of a joint distribution For a joint distribution F , the approximation ratio of a mechanism M = (x, p) to the optimal welfare is OP T F MF .
For a joint distribution F , we denote by F s|b , the conditional cumulative distribution function of the seller, given that the buyer's value is b.Similarly, we denote by F b|s , the conditional cumulative distribution function of the buyer, given that the seller's value is s.We denote by ½ A the indicator function for the event A. For example, we will use ½ [s>b] to denote the indicator function for the event that the value of the seller s is larger than the value of the buyer b.
In this paper, we consider several notions of incentive compatibility of a mechanism M = (x, p).We start by defining incentive compatibility for only one of the players: • Seller Dominant Strategy Incentive Compatibility: for every s, s ′ , b: • Seller Bayesian Incentive Compatibility: for every s, s ′ , b: • Buyer Dominant Strategy Incentive Compatibility: for every b, b ′ , s: • Buyer Bayesian Incentive Compatibility: for every b, b ′ , s: We will say that a mechanism is Dominant Strategy (Bayesian) Incentive Compatible if it is dominant strategy (Bayesian) incentive compatible for both the buyer and the seller.A mechanism is one-sided dominant strategy incentive compatible if it is dominant strategy incentive compatible for at least one of the players and Bayesian incentive compatible for the other.
A distribution F b|s is equal revenue distribution for a seller with value s, if, for every value of p in the support of the buyer's distribution F b|s , the expected payment to the seller from a take-it-or-leave-it offer of price p to the buyer is the same.
For example, consider a seller with a value 0, and a buyer with distribution F b|0 (b): The expected payment to the seller from any take-it-or-leave-it offer of price 1 ≥ p to the buyer is 1.
The expected profit of a buyer with value b, from a take-it-or-leave-it offer of price p to the seller is A distribution F s|b is equal profit distribution for a buyer with value b, if, for every value of p in the support of the seller's distribution F s|b , the expected profit of the buyer from a take-it-or-leave-it offer of price p to the seller is the same.For example, consider a buyer with value 1, and a seller with distribution F s|1 (s): The expected profit of the buyer from any take-it-or-leave-it offer of price p ∈ [0, 1 − 1 e ] to the seller is 1 e .In the buyer-offering mechanism, a buyer with value b makes a take-it-or-leave-it offer of price p to the seller, where the price p maximizes the buyer's profit, i.e., maximizes (b − p) • F s|b (p).
3 An e e−1 -Approximation for Correlated Values In this section, we prove that the buyer-offering mechanism provides an e e−1 ≈ 1.58 approximation to the optimal welfare, even if the values are drawn from a joint distribution.Recall that even when the values are drawn from independent distributions, the best currently known approximation mechanisms achieve a close approximation ratio of ≈ 1.38 [CW23,LRW23].Our approximation guarantee holds for all distributions for which the buyer-offering mechanism is well defined (i.e., there always exists an offer that maximizes the profit), but note that there are distributions for which the buyer-offering mechanism is not well defined.For example, consider the following joint distribution F ε for some small 0 < ε < 1.In F ε , the buyer has only one value 1, and the seller's value is supported on the interval (0, e−ε−1 e−ε ], with marginal distribution function F ε s : Let f (p) = (1 − p) • F ε s (p) be a function that denotes the expected profit of the buyer from a take-it-or-leave-it offer of p to the seller.Then, f (0) = 0, and for every p ∈ (0, e−ε−1 e−ε ], we get f (p) = 1+ε(1−p) e .Observe that the derivative of f for values p ∈ (0, e−ε−1 e−ε ] is −ε e , which is negative.Hence, the buyer's expected profit from a take-it-or-leave-it offer of p to the seller is a strictly decreasing function in the interval p ∈ (0, e−ε−1 e−ε ].Since the interval is open at 0, the function does not have a maximum.Thus, the buyer-offering mechanismis not defined for this distribution.We prove that the buyer-offering mechanism provides an e e−1 -approximation for all distributions for which it is defined.For the remaining distributions, we show that a slight variant of the buyer-offering mechanism provides a similar approximation ratio: Theorem 3.1. 1.For every joint distribution F of the values of the buyer and seller, the buyer-offering mechanism provides an e e−1 approximation to the optimal welfare if the buyer-offering mechanism is well-defined.2. For every joint distribution F of the values of the buyer and seller and every ε > 0, there is a one-sided dominant strategy mechanism that provides an e e−1 + O(ε) approximation to the optimal welfare.
We first prove the first part of the theorem.We then use the first part to prove the second part.After establishing the theorem, we discuss extending the result to double auctions (Subsection 3.4).

Proof of Theorem 3.1: Part I
Fix a value b of the buyer in the support of F .Denote by F s|b the distribution of the seller's value given that the value of the buyer is b.We will show that, for every b, the approximation ratio of the buyer-offering mechanism when the value of the buyer is always b and the value of the seller is always F s|b is e e−1 .This immediately implies that the approximation ratio of the buyer-offering mechanism when the values are drawn from F is e e−1 .Let p b be the price that the buyer offers when his value is b.Since the value b is fixed, to simplify notation, we drop the subscripts from p b and F s|b , and simply write p and F instead.
We now bound from above the approximation ratio of the buyer-offering for the distribution F (i.e., the expected optimal welfare divided by the expected welfare of the buyer-offering mechanism).
Let q 1 = F (p) and let q 2 = F (b).We have that q 1 ≤ q 2 (since p ≤ b).Observe that if q 2 = q 1 , the approximation ratio is 1, as needed.Therefore, we assume that q 1 < q 2 .Rewriting the RHS of (1) we have: For fixed q 1 , q 2 and b, this expression is maximized when E s∼F [s|p < s ≤ b] • (q 2 − q 1 ) is minimized.Therefore, in Lemma 3.2, we bound from below the expression E s∼F [s|p < s ≤ b] • (q 2 − q 1 ) to achieve an upper bound on the approximation ratio.
Recall that the price p maximizes the buyer's profit for the distribution F .We use this to bound from above F (p ′ ) for p < p ′ < b.Let u = q 1 • (b − p) be the expected profit of the buyer when the price is p.Then: For p ′ = b − u q2 , we get that the bound (3) is tight and equal to q 2 6 .Let f be the seller's probability density function given that the buyer's value is b.We want to bound from below the expression: We use the following bounds on F (s): for p ≤ s ≤ b − u q2 we have F (s) ≤ u b−s (by Inequality (3)) and for b − u q2 ≤ s ≤ b we have F (s) ≤ q 2 .Now, Overall, we get that the approximation ratio is bounded from above by

Proof of Theorem 3.1: Part II
We consider a slight variant of the buyer-offering mechanism.Given ε > 0, set δ = ε • OP T , where OP T is the expected value of the optimal social welfare.The ε-buyer offering mechanism makes a profit-maximizing take-it-or-leave-it offer p for the seller, where p belongs to the set of offers that are a multiple of δ.I.e., p = k • δ for some k ∈ N.This mechanism is Bayesian incentive compatible since the set of possible prices for a buyer with value b contains at most ⌈ b k ⌉ elements.Since this set is finite for every value b, it has a maximum-profit element.In addition, the seller obviously has a dominant strategy.
As in the first part, we prove the approximation guarantee for every value b of the buyer.Let F s (s) be the marginal distribution of the seller given b.Obtain a "discretized" F ′ s (s) by "pushing" the mass of F s (s) in all points that are not a multiple of δ to the nearest (from above) multiple of δ.Note that the buyer offering mechanism is now defined since the support of F ′ s (s) is finite.Thus, there is an offer with a maximum profit since for every offer that is not in the support there is an offer in the support with at least the same profit.Furthermore, observe that for every b, if the buyer offering mechanism makes an offer p when the marginal distribution is F ′ s (s) then p is also the offer that ε-buyer offering mechanism makes when the marginal distribution is F s (s).Observe that F s (s) and F ′ s (s) are very close to each other, and thus the expected welfare that the buyer offering mechanism provides for F ′ s (s) and the expected welfare of the ε-buyer offering mechanism for F s (s) differ only by δ.The second part of the theorem now follows since, by the first part, the buyer-offering mechanism provides an e e−1 approximation for F ′ s (s) and because the optimal welfare of F ′ s (s) and the optimal welfare of F s (s) differ only by δ.

Tightness of Analysis
We now present an instance of a distribution F where the buyer-offering mechanism yields an approximation ratio no better than e e−1 to the optimal welfare.Subsequently, in Subsection 4, we establish a more robust result that states that no one-sided dominant strategy mechanism can offer an approximation ratio better than e e−1 .Since the buyer-offering mechanism is a one-sided dominant strategy mechanism, the result of Subsection 4 also implies that the analysis of the buyer-offering mechanism is precise.Nevertheless, we provide a direct analysis of the buyer-offering mechanism in this section for simplicity.
Let F be a joint distribution over the buyer and seller values in which the value of the buyer is always 1, and the value of the seller is in [0, e−1 e ].The seller's value is distributed as follows: s > e−1 e .
7 Recall that q 1 ≤ q 2 , and so 0 ≤ q 1 q 2 ≤ 1.Thus, the function f (x) = 1 1+x ln x is maximized when x = 1 e , and its maximal value is e e−1 .
Note that the seller's distribution is an equal profit distribution for a buyer with value of 1. I.e., for the buyer, every price p ∈ [0, e−1 e ] yields the same expected profit.Thus, by tie-breaking, we may assume that the buyer-offering mechanism uses price 0 (alternatively, to eliminate the use of tie-breaking, one can change F s and slightly increase the probability that the seller's value is 0 -the analysis remains almost identical).
We now analyze the approximation ratio of the buyer-offering mechanism for the distribution F :

Double Auctions
In double auctions there are multiple buyers and sellers, each seller i owns a single item and his value for it is s i , all items are identical, and each buyer j wants one unit and his value for it is b j .Similarly, to bilateral trade, we wish to approximate the optimal welfare, which in the double auction case with k sellers is equal to the sum of the k largest values among the sellers and buyers.
Our positive result for bilateral trade also implies a positive result for double auctions.Similarly to the work of Babaioff et al. [BCGZ18], to obtain a solution for the double auction case, we combine McAfee's trade reduction mechanism [McA92] with the mechanism we use for the bilateral trade case (the buyer offering) in the following manner: compute the maximal number of efficient trades (where trade is efficient only if the value of the buyer is larger than the value of the seller).Run the trade reduction mechanism if there are at least two efficient trades.If there is only one efficient trade, run the buyer offering mechanism for the bilateral trade problem with the buyer being the highest-value buyer and the seller being the lowest-value seller.The distribution over the seller's value is the conditional distribution over the value of this seller given all values except his own and that he is the lowest value seller and the additional requirement that the price must be at least as large as the value of the second highest buyer.
Similarly to [BCGZ18], this mechanism is Bayesian incentive compatible and ex-post individually rational.Observe that the approximation ratio of this mechanism is at least 2. If there are at least two efficient trades, the approximation ratio of the trade reduction mechanism is 1 − 1 k where k is the number of efficient trades, which gives us a 2 approximation guarantee.If there is only one efficient trade, we get an approximation guarantee that is at least as good as the approximation guarantee of the buyer-offering mechanism for the bilateral trade case, which is e e−1 .Lastly, if the number of efficient trades is 0, we do nothing and get an optimal approximation.

The Limits of One-Sided Dominant Strategy Mechanisms
We now prove that no one-sided dominant strategy mechanism can provide an approximation ratio better than e e−1 .This shows, in particular, the optimality of the buyer-offering mechanism as its approximation ratio is e e−1 (Theorem 3.1).It also shows, for example, that taking the better of the buyer-pricing mechanism and a fixed price mechanism does not improve the approximation ratio.In Subsection 4.1, we present a specific distribution for which every one-sided dominant strategy mechanism does not provide an approximation ratio better than e e−1 .We provide the formal analysis of the impossibility in Subsection 4.2.

The Distribution F k,ε
For k ∈ N, ε > 0, let F k,ε be a joint distribution over the buyer and seller values in which the buyer receives values in [1, k] ∪ {k + 1} with probability density function of , the seller's value is distributed according to an (almost) equal profit distribution of the buyer.I.e., when the buyer's value is b, the cumulative distribution function of the value of the seller is F k,ε s|b : Note that the buyer's marginal distribution always sums up to 1: Moreover, for every value in the buyer's support, the conditional distribution of the seller sums up to 1: e +ε)+ε = 1.This distribution has two useful properties.The first is that when the seller's value is 0, the buyer's distribution is very close to an equal revenue distribution (it is implied by the marginal probability density function of the buyer that is close to 1 b 2 ).The second property is that for every value b in the buyer's support, the seller's distribution is very close to an equal profit distribution.
Fix some mechanism M that is one-sided dominant strategy for one of the players.If M is dominant strategy for the buyer, our bound on the welfare is achieved by utilizing the first property.In this case, only the seller's value can affect the value of the offer.Given that the seller's value is 0, the buyer's distribution is very close to an equal revenue distribution.We show that when the seller's value is 0, the seller will strictly prefer higher prices.Since the mechanism is Bayesian incentive compatible for the seller, the highest take-itor-leave it offer is when s = 0. Very roughly speaking, this implies that the welfare of the mechanism is low: if the highest offer is low, then the contribution to the welfare of trades when s = 0 is high, but no trade is done for larger values of the seller, which happens with significant probability.On the other hand, if the value of the highest offer p is large, trade is less likely in the s = 0 case.
If M is dominant strategy for the seller, we utilize the second property.Now, only the buyer's value can affect the offer price.However, the buyer strictly prefers lower offers as lower prices will yield higher profit.Since the mechanism is incentive compatible for the buyer, the offer price is the same for every value b of the buyer (otherwise, the buyer will prefer the lower offer and deviate from his equilibrium strategy).Then, every mechanism for F k,ε that offers a take-it-or-leave-it-offer p to the seller (one-sided dominant strategy for the buyer) is a fixed price mechanism.It only remains to show that a fixed price mechanism has low welfare.

Analysis of One-Sided Mechanisms for the Distribution F k,ε
Theorem 4.1.Let k ≥ 2 and ε > 0. Every one-sided dominant strategy mechanism for F k,ε provides an approximation ratio of at least e e−1 as ε approaches 0 and k approaches ∞.To prove this theorem we use the family of allocation rules x p (Definition 4.2).We show that the welfare of every one-sided dominant strategy mechanism for the seller is at most the welfare of x p , for some p (Claim 4.4).Similarly, we show that the welfare of every one-sided dominant strategy mechanism for the buyer is no better than the welfare of x p , for some p (Claim 4.3).We conclude the proof of the theorem by bounding the approximation ratio of every possible allocation rule x p (Lemma 4.5).Definition 4.2.For every p ≥ 0, let x p be the following allocation rule for the distribution F k,ε : Observe that for some values of p, this allocation rule x p , is not implementable.It is used simply to bound the welfare of every one-sided dominant strategy mechanism.The next three claims suffice to prove the theorem.Their proofs can be found in Subsections 4.2.1, 4.2.2, and 4.2.3.Claim 4.3.Let k ≥ 2 and ε > 0. Let M be a one-sided dominant strategy mechanism for the buyer.There exists p ≥ 0, such that the welfare of M is at most the welfare of x p , both with respect to the distribution F k,ε .Claim 4.4.Let k ≥ 2 and ε > 0. Let M be a one-sided dominant strategy mechanism M for the seller.There exists p ≥ 0, such that the welfare of M is at most the welfare of x p , both with respect to the distribution F k,ε .Claim 4.5.Fix p ≥ 0. When ε approaches 0 and k approaches ∞, the allocation rule x p provides an approximation ratio no better than e e−1 to the optimal welfare for the distribution F k,ε .
4.2.1 Mechanisms with Dominant Strategy for the Buyer (Proof of Claim 4.3) Every one-sided dominant strategy mechanism for the buyer is a take-it-or-leave-it offer for the buyer, where the price may only depend on the seller's value.Fix such a mechanism M = (x, p) for the distribution F k,ε .For every value s in the seller's support, denote the take-it-or-leave-it offer of the mechanism M by p s .Now, by Lemma 4.6, the price p 0 offered to the buyer when the seller's value is 0, should be no lower than any price in {p s |s ∈ [0, (k + 1) • e−1 e + ε]}.This is true since if there is a price p ′ s > p 0 , and the seller prefers higher prices for p 0 (Lemma 4.6), the seller will play the strategy that sets the price to p ′ s , in contradiction to the incentive compatibility of the mechanism.Recall that the item can be traded only if the seller's value is at most the price.Thus, the mechanism M can only trade when s < p s ≤ p 0 .Now, let p = p 0 , and consider the allocation rule x p (Definition 4.2).Observe that the allocation rule x p trades the item in every instance that M trades the item and might trade the item when M does not.Therefore, the welfare of x p is at least the welfare of M .Lemma 4.6.For s = 0 and p ∈ [1, k], the expected profit of the seller from a take-it-or-leave-it offer with price p to the buyer is higher than a take-it-or-leave-it offer with price p ′ < p.
Proof of Lemma 4.6.Recall that the conditional cumulative distribution function of the buyer, given that the seller's value is 0 is denoted by F k,ε b|0 .The profit of the seller with value 0 from a take-it-or-leave-it offer of To analyze this expression, we first provide an explicit expression for By definition, we have: . We now prove that g is strictly increasing by showing that its derivative is positive.We get that the profit of the seller with value 0 from a take-it-or-leave-it where in the last inequality we assume that ε − ε 2 > 0, since we can choose ε > 0 to be as small as we want.We prove that ε+2p 2(p+ε) 2 − p 2 (ε+p) 3 > 0, which implies that g ′ (p) > 0 for every p ∈ [1, k]: This proves that the expected profit of the seller with value 0 from a take-it-or-leave-it offer of p ∈ [1, k] is smaller than his expected profit from an offer of p ′ ∈ [1, k] that is smaller than p. Observe that the expected profit of the seller with value 0 from a take-it-or-leave-it offer of p < 1 is even smaller than his expected profit from an offer of price 1 (as reducing the offer below 1, does not increase the probability that the buyer will buy the item).The profit of an offer p = k + 1 is even larger profit than an offer of k, as the probability 1 − F k,ε b|s=0 (k + 1) is 1 − F k,ε b|s=0 (k) but the price is larger (k + 1 > k).This concludes the proof of the lemma.

Mechanisms with Dominant Strategy for the Seller (Proof of Claim 4.4)
Every one-sided dominant strategy mechanism for the seller is a take-it-or-leave-it offer to the seller, where the offer depends only on the value of the buyer.Fix a mechanism M = (x, p) for the distribution F k,ε .For every b ∈ [1, k] ∪ {k + 1}, the mechanism offers a price p b .By Lemma 4.7, the expected profit of the buyer with value b ∈ [1, k] ∪ {k + 1} is higher when p b is smaller.Since the mechanism is Bayesian incentive compatible for the buyer we claim that it must be a fixed price, i.e., p b is equal for every b ∈ [1, k] ∪ {k + 1}.This is true since if there are two different values p b > p b ′ and the buyer strictly prefers lower prices (Lemma 4.7), the buyer will play the strategy that sets the price to p b ′ , in contradiction to the incentive compatibility of the mechanism.Let p be the fixed price and consider the allocation rule x p (see Definition 4.2).Recall that the item can be traded only if the seller's value is at most the price.Thus, the mechanism M only trades the item when s ≤ p.
Observe that the allocation rule x p trades the item in every instance that M trades the item and might trade the item when M does not.Thus, the welfare of x p is at least the welfare of M .
, the expected profit of the buyer from a take-itor-leave-it offer with price p to the seller is higher than a take-it-or-leave-it offer with price p ′ > p. s|b is a slightly skewed equal profit distribution, lower prices yield strictly higher profits, and so g is a strictly decreasing function.Formally, g(p) is a decreasing function since its derivative is negative for every value p in its range: 4.2.3Bounding the Approximation Ratio (Proof of Claim 4.5) We start with computing the optimal welfare of the distribution F k,ε .Observe that in every instance in the distribution F k,ε , the buyer has a higher value than the seller, thus trade occurs in the optimal allocation rule.The optimal welfare is therefore: This expression approaches ln k + 1 + 1 k as ε approaches 0. We consider the possible values of the buyer's value and compute the contribution of each value to the expected welfare when the item is traded according to x p .When b = k + 1, x p always sells the item (as does the optimal allocation rule).This contributes (k + 1) 1 k+ε + ε 1+ε to the expected welfare.
For every b ∈ [1, k], with probability b e(b+ε) , the seller's value is 0.Then, according to x p , the item is traded only when b ≥ p.These instances contribute and the seller's value is larger than 0, the item is traded only when s ≤ p according to x p .These instances contribute ] db to the expected welfare.Thus, the expected welfare is the sum of these three expressions: Next, we consider each expression separately, bound it from above, and take its limit as ε goes to 0. We start with the first expression (4): We continue with the second expression (5).By the dominated convergence theorem, we can swap the order of the integral and the limit operator 8 .lim Finally, for the third expression (6), we first break it into three expressions: Then, we bound each expression and take its limit as ε approaches 0. Again, by the dominated convergence theorem, we can swap the order of the integral and the limit operator 8 . .Now, by summing all the expressions together, we get that as ε approaches 0, the welfare of x p approaches: 1 Thus, as ε approaches 0, the approximation ratio approaches As k approaches ∞, the upper bound approaches e e−1 .

Impossibilities for Bayesian Mechanisms
In this section, we prove several impossibility results for Bayesian incentive-compatible mechanisms.Specifically, in Subsection 5.2, we prove a lower bound of 1.113 for the welfare of independent distributions; in Subsection 5.4 we prove a lower bound of 1 + ln 2 2 for the welfare of correlated distributions, and in Subsection 5.5 we prove a lower bound of 2 for the gains from trade of independent distributions.
For the welfare lower bounds (Subsection 5.2, and Subsection 5.4), the approximation ratio is better than the ratio of 1.071 that is obtained by Blumrosen and Mizrahi [BM16].In the gains from trade lower bound (Subsection 5.5), the approximation ratio is better than the ratio of e 2 ≈ 1.359, also in [BM16].However, our results only apply to deterministic Bayesian mechanisms, whereas their results apply to randomized ones.We note that all major mechanisms considered in the literature (e.g., fixed price auction, buyer and seller offering mechanisms) are deterministic 9 .
In the proofs, we rely on representing joint distributions and allocation rules in tables notation.Thus, this section starts with Subsection 5.1, which explains this notation.The proofs of Section 5.4 and Section 5.5 share a similar structure.Both rely on L-shaped distributions, presented and analyzed in Section 5.3.

Preparations: Representing Distributions and Mechanisms
We consider only discrete distributions in this section (since this section aims to prove impossibilities, this only strengthens our results).We represent a discrete (joint) distribution using tables: each row corresponds to one of the possible values of the buyer, and each column corresponds to one of the possible values of the seller.Each cell (s, b) corresponds to an instance in the support of the distribution.The value in the cell (s, b) is the probability the instance (s, b) is realized.We represent an allocation rule for the distribution by adding * to a cell (s, b) if the item is traded in the instance (s, b).
Example 5.1.An illustration of a distribution where both the seller and the buyer have support of size 2. 9 Our impossibilities also apply to mechanisms which are a probability distribution over deterministic ex-post individually rational mechanisms, like the random offerer mechanism [DMSW22].To see this, consider such a mechanism A. A simple averaging argument shows that in the support of A, there must be a (deterministic and ex-post individually rational) mechanism A ′ that its approximation ratio is at least the approximation ratio of A. Our impossibilities directly apply to A ′ , and the approximation ratio of A is no better than the approximation ratio of A ′ .
Figure 1: An illustration of a distribution and an allocation function.Here, the buyer has two possible values b 1 < b 2 , and the seller also has two possible values s 1 < s 2 .The probability of the instance (s 1 , b 1 ) is x 1 , the probability of the instance (s 2 , b 1 ) is x 2 , the probability of the instance (s 1 , b 2 ) is x 3 , and the probability of the instance (s 2 , b 2 ) is x 4 (x 1 + x 2 + x 2 + x 4 = 1).The item is traded in all instances except (s 2 , b 1 ), as is indicated by the symbol * added to a cell if the item is traded in the instance that corresponds to that cell.

Warm-Up: Impossibilities for Welfare Approximation via 2 × 2 Distributions
We now prove an impossibility result of Bayesian incentive-compatible mechanisms for independent distribution, achieved by distributions where each player has only two possible values.Consider the family of distributions depicted in Figure 2. In every distribution that belongs to the family, the values of the buyer and the seller are independent.The buyer's value is 1 with probability x 1 and is b 2 with probability x 2 = 1 − x 1 .The seller's value is 0 with probability q 1 and is s 2 with probability q 2 = 1 − q 1 .We have that b 2 > s 2 > 1 > 0. The welfare maximizing allocation function of such distributions is depicted in Figure 2a.Every such distribution is called simple.
Theorem 5.2.There is a simple distribution F such that no Bayesian incentive compatible mechanism provides an approximation ratio better than 1.113 for the distribution F .proving this theorem, fix a simple distribution F .We now consider the possible allocation functions for F .Since F has 4 instances in its support, there are 2 4 = 16 possible allocation functions.However, in one of these instances (s 2 , b 1 ), trade cannot occur since b 1 < s 2 , resulting in 2 3 = 8 possible allocation functions.Observe that any of the four allocation functions that trade the item in at most one instance will result in lower welfare compared to at least one of the allocation functions shown in Figures 2c and 2d.Specifically, the allocation functions in figures 3a and 3b have lower welfare than the allocation in Figure 2c, and the allocation functions in figures 3c and 3d have lower welfare than the allocation in Figure 2d.Therefore, it is sufficient to obtain a lower bound on the approximation ratio of the allocation functions shown in Figures 2c and 2d to obtain a lower bound on the approximation of all the allocation functions shown in Figure 3.
Hence, from now on, we consider only the four allocation functions depicted in Figure 2, as the welfare of any other implementable allocation function for F is at most the welfare of one of those four functions.
We choose the values of b 2 , s 2 , x 1 , x 2 , q 1 , q 2 so that the welfare maximizing allocation function, depicted in Figure 2a, is not implementable by a Bayesian incentive compatible mechanism (Lemma 5.3).Our choice of values also rules out the existence of a Bayesian implementation of the allocation rule in Figure 2b (Lemma 5.4).Note that the allocation rules in which the item is traded only when b = b 2 (depicted in Figure 2d) or only when s = 0 (depicted in Figure 2c) are always implementable.In fact, they are implementable by a fixed price mechanism: the former by fixed price of b 2 (or s 2 ), and the latter by fixed price of 0 (or 1).
Hence, we are left with the allocation functions depicted in Figure 2c, Figure 2d.We choose the values of the parameters such that the welfare of these two allocation functions will be equal (Lemma 5.5) and as low as possible.Together, we will get that for the distribution F , there is no Bayesian incentive compatible mechanism that provides an approximation ratio better than 1.113 (Subsection 5.2.1).
Figure 3: All allocation functions of F that trade the item in at most one instance.
, then the welfare-maximizing allocation function (depicted in Figure 2a) is not implementable by a Bayesian incentive compatible mechanism.
Proof of Lemma 5.3.Let M = (x, p) be a mechanism for F that implements the welfare-maximizing allocation function.Let p = p(0, b 2 ).The expected profit of the seller when his value is 0 and he follows his equilibrium strategy is x 1 • p(0, 1) + x 2 • p.The expected profit of the seller when his value is 0 and he plays the equilibrium strategy of s 2 is x 2 •p(s 2 , b 2 ).The mechanism is incentive compatible for the seller, and thus x 1 •p(0, 1)+x 2 •p ≥ x 2 • p(s 2 , b 2 ) (Inequality (2), for s = 0 and s ′ = s 2 ,).By individual rationality, the price is at most the value of the buyer p(0, 1) ≤ 1.Also, the price is at least the value of the seller p(s 2 , b 2 ) ≥ s 2 .Together, we get: Similarly, by Bayesian incentive compatibility for the buyer (Inequality (2) with b = b 2 and b ′ = 1): Inequalities ( 7) and (8) imply that ) + 1 then the allocation rule depicted in Figure 2b is not implementable by a Bayesian incentive compatible mechanism.
Proof of Lemma 5.3.Let M = (x, p) be a Bayesian incentive compatible mechanism that implements the allocation rule depicted in Figure 2b.
Together with the assumption and this is a contradiction as b 2 > s 2 , and thus the allocation rule depicted in Figure 2b is not implementable by a Bayesian incentive compatible mechanism.
Lemma 5.5.For x 1 • q 1 = x 2 • q 2 (b 2 − s 2 ), the expected welfare of the allocation rules of Figure 2c and in Figure 2d are equal.
Proof of Lemma 5.5.The expected welfare of the allocation rule in Figure 2c is: The expected welfare of the allocation rule in Figure 9 is: The two equations are equal when

Concluding the Proof of Theorem 5.2
We choose values for b 2 , s 2 , x 1 , x 2 , q 1 , q 2 so that the conditions of Lemmas 5.3, 5.4, and 5.5 are met.If the conditions hold, by these lemmas, the approximation ratio of every Bayesian incentive compatible mechanism for F is at least the approximation ratio of the mechanism depicted in Figure 2d.We, therefore, choose the values for the parameters so that this approximation ratio is maximized, i.e., this expression is maximized:

L-Shaped Distributions
We now define and analyze L-shaped distributions, which are used in Section 5.4 and Section 5.5.The heart of the proofs in these sections is Claim 5.8.In this section, we define and analyze the basic properties of L-shaped distributions and their standard form (Section 5.3.2) and prove Claim 5.8.Consider the family of L-shaped distributions illustrated in Figure 4.In each distribution in this family, the size of the support of the buyer's and seller's values is k.The buyer's support is b This family of distribution is called L-shaped as every distribution in this family satisfies a condition that restricts the trade only to instances inside the L-shape.Consider the set of instances that are outside the L-shape, i.e., L = {(s j , b i )|s j = 0 and b i = b k }.Then, every distribution in the family either has 0 probability for every instance in L, or in every instance in L, the seller's value is larger than the buyer's value.Either way, a trade cannot occur in the instances in L.
We consider L-shaped distributions as they are easier to analyze.Every allocation rule for an L-shaped distribution is associated with two threshold values: b i ∈ {b 1 , . . .b k } ∪ {∞} and s j ∈ {s 2 , . . ., s k } ∪ {0} (Observation 5.6).The allocation in the instances {(0, 1), (0, b 2 ), . . ., (0, b k−1 )} is determined by the threshold b i : if the buyer's value is below b i , the item is not traded; if the buyer's value is at least b i , the item is traded.Similarly, the allocation in the instances {(s 2 , b k ), . . ., (s k , b k )} is determined by the threshold s j : if the seller's value is above s j , the item is not traded; if the seller's value is at most s j , the item is traded.This property reduces the number of potential allocation rules of Bayesian incentive-compatible mechanisms.Furthermore, it provides a simple characterization of such allocation rules.Figure 4: An L-shaped distribution and the allocation function that maximizes its welfare (and its gains from trade).Each cell corresponding to an instance with positive probability is marked by +.A cell that corresponds to an instance that might have 0 probability is marked by ?.The cells where trade occurs in the optimal allocation rule are colored in orange .

Characterizing Implementable Allocation Functions for L-shaped Distributions
We now characterize the allocation rules that are implementable by Bayesian incentive-compatible mechanisms in L-shaped distributions.
Observation 5.6.Suppose M = (x, p) is a Bayesian incentive compatible mechanism for an L-shaped distribution F k .Then, there exist two threshold values b i ∈ {b 1 , . . .b k } ∪ {∞} and s j ∈ {s 2 , . . ., s k } ∪ {0} that determine the allocation in the instances allocation in the 1), (0, b 2 ), . . ., (0, b k−1 )} is determined by the threshold b i : if the buyer's value is below b i , the item is not traded; if the buyer's value is at least b i , the item is traded.Similarly, the allocation in the instances {(s 2 , b k ), . . ., (s k , b k )} is determined by the threshold s j : if the seller's value is above s j , the item is not traded; if the seller's value is at most s j , the item is traded.
By Observation 5.6, the only instance inside the L-shape whose allocation is not determined by the thresholds b t , s t is (0, b k ).Hence, each implementable allocation rule for an L-shaped distribution can be characterized by three parameters: b t , s t , and x(0, b k ).
Note that while every allocation rule that can be implemented by a Bayesian incentive-compatible mechanism can be described using these three parameters, not every allocation rule described by these parameters can be implemented by a Bayesian incentive compatible mechanism.
Proof of Observation 5.6.We prove the claims regarding the threshold for the buyer.The proof for the threshold of the seller is very similar.Consider an L-shaped distribution F k .Whenever the buyer's value is b ∈ {b 1 , . . .b k−1 }, a trade can only occur when the seller's value is 0. Let b i ∈ {b 1 , . . .b k−1 } be the smallest value in {b 1 , . . .b k−1 } such that the item is traded in the instance (0, b i ) by M .If the item is not traded in the instances {(0, 1), (0, b 2 ), . . ., (0, b k−1 )} we say that the threshold is ∞.
Assume that b i is not a threshold, i.e., there exists b j ∈ {b i+1 , . . ., b k−1 } such that in the instance (0, b j ), the item is not traded by M .The expected profit of the buyer when his value is b j and he follows his equilibrium strategy is Pr(s , which equals 0 as by individual rationality if the buyer does not get the item he pays nothing.The expected profit of the buyer when his value is b j and he plays the equilibrium strategy of b i is Pr Thus, the item is traded in the instance (0, b j ) by M .Furthermore, observe that it is not enough that the item is traded in (0, b j ); it should also be traded for the same price.Otherwise, for two values b, b ′ that are at least b i with p(0, b) < p(0, b ′ ), a buyer with value b ′ will prefer the equilibrium strategy of b.
Then, every allocation rule (b i , s j , 0) that trades the item in at least k instances is not implementable by a Bayesian incentive compatible mechanism.
Lemma 5.11.Suppose that for every 1 Then, all allocation rules (b t , s t , 1) that trade the item in exactly k instances have the same welfare and have the same gains from trade.
Observation 5.12.If Condition 1 of Definition 5.7 holds then so does the following condition: For every Proof of Claim 5.8.Consider an L-shaped distribution in a standard form.By the second condition (2) in Definition 5.7, and by Lemma 5.11, we have that every allocation rule (b t , s t , 1) that trades the item in exactly k instances have the same welfare and the same gains from trade, we denote this welfare by W k and this gains from trade by GF T k .We prove that if an allocation rule (b t , s t , x(0, b k is implementable, then its welfare is at most W k , and its gains from trade is at most GF T k . Let (b i , s j , 1) be an allocation rule that trades the item in the instance (0, b k ).If it trades the item in more than k instances, then, by Condition 1 of Definition 5.7 and Lemma 5.9, it cannot be implemented by a Bayesian incentive-compatible mechanism.If it trades the item in exactly k instances, its welfare is W k , and its gains from trade is GF T k .However, if it trades the item in less than k instances, then we claim that its welfare and gains from trade are lower than those of some other allocation rule (b ′ i , s ′ j , 1), which trades the item in k instances.When b i = ∞, then the allocation rule (∞, s k , 1), trades the item in every instance that (b i , s j , 1) does and even in instances that it does not.Similarly, when s j = 0, then the allocation rule (b 1 , 0, 1), trades the item in every instance that (b i , s j , 1) does and even in instances that it does not.Finally, when both b i = ∞ and s j = 0, then the allocation rule (b i , s i , 1) trades the item in every instance that (b i , s j , 1) does and even in instances that it does not, since i > j, as the number of instances in which the allocation rule (b i , s i , 1) trades the item is k − i + j less than k.
Let (b i , s j , 0) be an allocation rule that does not trade the item in the instance (0, b k ).If it trades the item in at least k instances, then, by Condition 1 of Definition 5.7, Observation 5.12, and Lemma 5.10, it cannot be implemented by a Bayesian incentive compatible mechanism.However, if it trades the item in at most k − 1 instances, its welfare and gains from trade are lower than the welfare and gains from trade of the allocation rule (b i , s j , 1), which trades the item in at most k instances and has a welfare of at most W k and gains from trade of at most GF T k (as discussed above).Therefore, the welfare and gains from trade of any implementable allocation rule (b t , s t , x(0, b k )) is at most W k and at most GF T k , respectively.
Proof of Lemma 5.9.Observe that if an allocation rule (b i , s j , 1) trades the item in at least k + 1 instances, then b i ∈ {b 1 , . . ., b k−1 }, and s j ∈ {s 2 , . . ., s k }.If b i ∈ {b 1 , . . ., b k−1 } and s j ∈ {s 2 , . . ., s k }, then the allocation rule (b i , s j , 1) trades the item in exactly k − i + j instances.Finally, k − i + j ≥ k + 1 for every 1 ≤ i ≤ k − 1 and every i + 1 ≤ j < k.Hence, we prove that for every 1 ≤ i ≤ k − 1 and every i + 1 ≤ j < k, if: then the allocation rule (b i , s j , 1) is not implementable by a Bayesian incentive compatible mechanism.Fix 1 ≤ i ≤ k − 1 and i + 1 ≤ j < k and consider the allocation rule (b i , s j , 1) (see Figure 5).Proof of Lemma 5.11.Note that an allocation rule (b i , s j , 1) trades the item in k −i−j instances if b i < ∞ and s j > 0, in j instances if b i = ∞, and in k − i + 1 instances if s j = 0.If both b i = ∞ and s j = 0, the allocation rule trades the item once.Therefore, the allocation rules that sell the item exactly k times are (b i , s i , 1) for 2 ≤ i ≤ k − 1, (∞, s k , 1), and (b 1 , 0, 1).To ensure that all of these allocation rules have the same welfare and same gains from trade, we require that each allocation rule (b i , s i , 1) has the same welfare and same gains from trade as its adjacent allocation rule.Specifically, for 2 ≤ i ≤ k − 2, the adjacent allocation rule is (b i+1 , s i+1 , 1) (see Figure 6 for an example).For (b k−1 , s k−1 , 1), the adjacent allocation rule is (∞, s k , 1).And for (b 1 , 0, 1), the adjacent allocation rule is (b 2 , s 2 , 1).6: The allocation rule (b i , s i , 1) and its adjacent allocation rule (b i+1 , s i+1 , 1) as defined in the proof of Lemma 5.11.The allocations according to (b i , s i , 1) are marked by * , and the allocations according to (b i+1 , s i+1 , 1) are marked by @.Each cell that corresponds to an instance with positive probability is marked by +, and a cell that corresponds to an instance that might have 0 probability and might not is marked by ?.
Note that for 2 ≤ i ≤ k − 2, an allocation rule (b i , s i , 1) has the same welfare and the same gains from trade as its adjacent allocation rule (b i+1 , s i+1 , 1) if the following equality holds: Similarly, for the allocation rule (b k−1 , s k−1 , 1) to have the same welfare and same gains from trade as its adjacent allocation rule (∞, s k , 1), we need to satisfy the equation: For the allocation rule (b 1 , 0, 1) to have the same welfare and the same gains from trade as its adjacent allocation rule (b 2 , s 2 , 1), we need the following equation to hold: Together, for every 1 Proof of Observation 5.12.Assume that Condition 1 of Definition 5.7 holds.For every 1 ≤ i ≤ k − 1 and i + 1 ≤ j < k: Observe that: and so:

Approximating the Welfare with Correlated Values
In this section, we prove a lower bound for approximating the welfare.We consider a family of correlated distributions that are L-shaped (Section 5.3).In every distribution F k (depicted in Figure 7) in the family, the number of instances in the support is only 2k − 1.When the buyer's value is b ∈ {1, b 2 , . . ., b k−1 }, then the seller's value is always 0. Similarly, when the seller's value is s ∈ {s 2 , . . ., s k }, then the buyer's value is always b k .The probability of the buyer's value being b i is When the buyer's value is b k , the probability that the seller's value is s i is q i , for i ∈ [k].In every instance in its support, the buyer's value is larger than the seller's value (i.e., s k < b k ), and so a trade improves the welfare in each instance in the support.Theorem 5.13.Let k ≥ 2. There exists an L-shaped distribution F k such that every Bayesian incentive compatible mechanism for F k provides an approximation ratio no better than 1 to the optimal welfare, where 1 k is the Harmonic number.This ratio approaches 1 + ln 2 2 as k approaches ∞.We choose values for the parameters of the distribution F k such that F k is in standard L-shaped form (Definition 5.7).Let ε > 0 be a small enough value, we set the values: (9) Next, we select the values for s 2 , . . .s k such that Condition 2 in Definition 5.7 is met, for every 1 ≤ i ≤ k − 1: Lemma 5.14.For the choice of parameters in Equation 9 and Equation 10, the first condition of Definition 5.7 holds.
By Lemma 5.15, our choice of values indeed yields a legal distribution, and the order over the values is as assumed throughout the proof, i.e., b We will prove these lemmas after using them to bound the approximation ratio .
As ε approaches 0, the bound where γ is Euler's constant.This proves the theorem assuming Lemmas 5.14 and 5.15.We now provide the proofs of these lemmas.
Proof of Lemma 5.14.The first condition of Definition 4 holds for every 1 ≤ i ≤ k − 1 and i + 1 ≤ j < k if: Then, it is enough to show that for every 1 ≤ i ≤ k − 1 and i + 1 ≤ j < k, the two inequalities hold: We start with the first Inequality 11a using our values for the parameters (Equation 9 and Equation 10: We are left with proving Inequality (11b).By recalling that j ≥ i + 1 ≥ 2 and our choice of values (Equations ( 9) and (10)): Proof of Lemma 5.15.We first verify that the buyer's marginal probability distribution sums up to 1: Moreover, for every value in the buyer's support, the conditional distribution of the seller also sums up to 1.For b ∈ {b 1 , . . ., b k−1 }, the seller's value is 0 with probability 1 and for b = b k , the sum of the seller's conditional probabilities is and for k ≥ 2, the last inequality holds.For i = k − 1, we get: .
where the last inequality holds for k ≥ 2. Next, we show that s i < s i+1 , for every 1 ≤ i ≤ k − 1.For i = 1, we have: and the inequality clearly holds.For 2 ≤ i ≤ k − 1, we get: Finally, we are left with proving s k < b k : and the last inequality clearly holds, as all values are strictly positive.

Approximating the Gains from Trade with Independent Values
This section proves a lower bound for approximating the gains from trade.We consider a family of independent distributions that are L-shaped (Section 5.3).In every distribution F k (depicted in Figure 8) in the family, the buyer's support is b The probability of the buyer's value being b i is x i for i ∈ [k], and the probability that the seller's value is In every instance that is not in the "L", i.e., in every instance in the set L = {(s j , b i )|s j = 0 and b i = b k }, the seller's value is is larger than the buyer's value ( s 2 > b k−1 ) and trade cannot occur.
Theorem 5.16.Let k ≥ 2. There exists an L-shaped distribution F k such that every Bayesian incentive compatible mechanism for F k provides an approximation ratio no better than 1 + H k −1 H k +1 to the optimal gains from trade, where H n = n k=1 1 k is the Harmonic number.This ratio approaches 2 as k approaches ∞.This ratio is tight for L-shaped distributions.To see this, consider two allocation rules: the first trades the item when s = 0, and the second trades the item when b = b k .Both allocation rules can be implemented by fixed price mechanisms: the first sets a price of 0 and the second a price of s k .The sum of the gains from trade of the two mechanisms is the optimal gains from trade, so at least one of them guarantees a ratio of 2.
Figure 8: An independent L-shaped distribution and the allocation function that maximizes its gains from trade.Thecolored cellsd in orange correspond to instances in which a trade can occur.
We choose values for the parameters of the distribution F k such that F k is in standard L-shaped form (Definition 5.7).Let ε > 0 be small enough.We set the values: Next, we select the values for s 2 , . . .s k such that Condition 2 in Definition 5.7 is met, for every 1 ≤ i ≤ k−1: Lemma 5.17.For the choice of parameters in Equations 12 and 13, the first condition of Definition 5.7 holds.
Lemma 5.18.For the choice of parameters in Equation 12 and Equation 13, we get a joint distribution over the values of the buyer and seller, and Observe that these two lemmas and our choice of parameters in Equation 13 imply a family of L-shaped distributions in standard form (Definition 4).Thus, we can apply Claim 5.8 and analyze the approximation ratio OP T F k GF T k .Recall that in an L-shaped distribution, the only instances in which trade can occur are in the "L".Thus, instances in the "L" are the only instances that affect the gains from trade.We now analyze the approximation ratio and then prove the two lemmas.
By Claim 5.8, the gains from trade of every mechanism (b i , s j , 1) that trades the item in k instances is the same, and we denoted it by GF T k .The allocation rule (b 1 , 0, 1) trades the item only in the instances where the seller's value is 0, and thus its gains from trade is q 1 k i=1 x i b i , and GF T k = q 1 k i=1 x i b i .
As ε approaches 0, the bound OP T F k GF T k approaches 1 + H k −1 H k −1+2 , and as k approaches ∞, OP T F k GF T k approaches 2, due to the fact that lim n→∞ (H n − ln n) = γ, where γ is Euler's constant.This proves the theorem assuming Lemmas 5.17 and 5.18.We now provide the proofs of these lemmas.
Proof of Lemma 5.17.The first condition of Definition 4 is that for every 1 ≤ i ≤ k −1 and every i+1 ≤ j < k: For our independent distributions, it is equal to: Then, it is enough to show that for every 1 ≤ i ≤ k − 1 and i + 1 ≤ j < k, the two inequalities hold: We start with the first Inequality 14a using our values for the parameters (Equation 12 and Equation 13): We now prove the second Inequality 14b, for every 2 ≤ i + 1 ≤ j ≤ k − 1: Proof of Lemma 5.18.We first verify that the buyer's marginal probability distribution sums up to 1: k i=1 x i = k−1 i=1 1 2 i + 1 2 k−1 = 1, and the seller's marginal distribution sums up to 1: for the values chosen in Equation 12.
Next, we verify that indeed b i < b i+1 , for every 1 ≤ i ≤ k − 1.For 1 ≤ i ≤ k − 2 we have: For i = k − 1, we get: Next, we show that s i < s i+1 , for every 1 ≤ i ≤ k − 1.For i = 1, we have: and the inequality clearly holds.For 2 ≤ i ≤ k − 1, we get: Next, we show that s 2 > b k−1 : Then it is enough to show that 3 2 > 1 k , which holds for k > 2 3 .Finally, we are left with proving s k < b k : and the last inequality clearly holds as k ≥ 1.
When the seller's value is 0 and he plays his equilibrium strategy, his profit is x 1 • r + x 2 • 2.33 = 1.57133, which is larger than his profit when he plays the equilibrium strategy of s 2 , x 2 • s 2 = 1.57036.Next, we show that for every value of the buyer in the support, the Bayesian incentive compatibility inequality (2) holds.When the buyer's value is 1 and he plays his equilibrium strategy his profit is q 1 (r −r) = 0 which is larger than his profit from playing the equilibrium strategy of b 2 , q 1 •(1−2.33)+q 2 •(1−s 2 ) < 0. When the buyer's value is b 2 and he plays his equilibrium strategy, his profit is q 1 (b 2 − 2.33) + q 2 (b 2 − s 2 ) > 4.2875, which is larger than his profit when he plays the equilibrium strategy of 1, q 1 • (b 2 • r − r) < 4.2867.Hence, M is Bayesian incentive compatible.M 's approximation ratio is: x 1 q 1 + x 1 q 2 s 2 + x 2 b 2 r • x 1 q 1 + x 1 q 2 s 2 + x 2 b 2 = 1 + (1 − r)x 1 q 1 r • x 1 q 1 + x 1 q 2 s 2 + x 2 b 2 < 1.00002.
Proof of Lemma B.3.The proof relies on Section 5.2.Recall that for the distribution of Section 5.2, there are 2 3 possible allocation functions (as in the instance (s 2 , 1) trade cannot occur since s 2 > 1).Observe that any of the four allocation functions that trade the item in at most one instance has smaller welfare compared to at least one of the allocation functions depicted in Figure 2c and Figure 2d.By Lemma 5.3, if s 2 − x1 x2 > q2 q1 (b 2 − s 2 ) + 1 then the welfare maximizing allocation function (depicted in Figure 2a) is not implementable by a Bayesian incentive compatible mechanism, and by our choice of parameters this is indeed the case: s 2 − x1 x2 > 2.3264, q2 q1 (b 2 − s 2 ) + 1 < 2.3253.Similarly, by Lemma 5.4, if s 2 − x1 x2 > q2 q1 (b 2 − s 2 ) + 1, then the allocation function depicted in Figure 2b is not implementable by a Bayesian incentive compatible mechanism, and by our choice of parameters this is indeed the case.Finally, the optimal approximation ratio of a deterministic Bayesian incentive compatible for F is the maximum between the approximation ratio of the allocation rule depicted in Figure 2d, and the approximation ratio of the mechanism depicted in Figure 2c.Next, we compute the two ratios.The approximation ratio of the allocation rule depicted in Figure 2d is: The approximation ratio of the allocation rule depicted in Figure 2c is: Hence, the approximation ratio of any deterministic Bayesian incentive compatible for F is at least 1.113.
Proof of Lemma 4.7.Let b ∈ [1, k] ∪ {k + 1}.The conditional cumulative distribution of the seller's value is F k,ε s|b .Now, for every p ∈ [0, b • e−1 e + ε], the expected profit of the buyer from a take-it-or-leave-it offer of price p to the seller is (b − p) • b e(b−p+ε) .We define the function g(p) = (b − p) • b e(b−p+ε) for every p ∈ [0, b • e−1 e + ε], and show that it is a strictly decreasing function.By definition, the function g(p) is the expected profit of the buyer from a take-it-or-leave-it offer of price p to the seller.Intuitively, if F k,ε s|b was exactly the equal profit distribution, i.e., F k,ε s|b (p) = b e(b−p) for every p ∈ [0, b • e−1 e + ε], then g was a constant function with value b e .However, since F k,ε − p) + ln (e − 1) − ln k + ln e e − 1 .

Figure 2 :
Figure 2: All possible allocation functions of F that trade the item in at least two instances.
which is grater than 0 as by individual rationality p(0, b i ) ≤ b i and since b j > b i .Note that incentive constraints are violated since Pr(s = 0| b = b Assume that this allocation rule is implementable by a Bayesian incentive compatible mechanism M = (x, p), and denote by p the trade price in the instance (0, b k ).By Bayesian incentive compatibility for the buyer (Inequality 2 with b = b k , b ′ = b i ) and the seller (Inequality 2 with s = 0, s ′ = s j ): b k • ( j r=1 Pr(s = s r |b = b k )) − j r=1 Pr(s = s r |b = b k )) • p(s r , b k ) ≥ Pr(s = 0|b = b k ) • (b k − p(0, b i )) Pr(b = b k |s = 0) • p(0, b k ) + k−1 l=i Pr(b = b l |s = 0) • p(0, b l ) ≥ Pr(b = b k |s = 0) • p(s j , b k ).

Figure 7 :
Figure 7: An L-shaped distribution F k and its welfare maximizing allocation function.The colored cells in orange correspond to instances with positive probability.
and s k < b k , Lemma 5.15.For the choice of parameters in Equation 9 and Equation 10, we get a joint distribution over the values of the buyer and seller, and b 1