Bilateral Trade with Correlated Values

TL;DR

The paper analyzes bilateral trade with correlated private values, introducing a buyer-offering mechanism that achieves a welfare approximation of $\frac{e}{e-1}$ under any joint distribution, with Bayesian incentive compatibility and a dominant strategy for the seller. It proves the optimality of this one-sided mechanism among all one-sided dominant-strategy mechanisms, and shows that no two-sided dominant-strategy mechanism can achieve a constant welfare approximation. It further establishes impossibility results for deterministic Bayesian mechanisms, including a tight bound of $1+\frac{\ln 2}{2}$ for welfare under correlated values and stronger bounds for gains-from-trade with independent distributions. The paper also extends the main welfare result to double auctions via a combination of McAfee’s trade reduction and the buyer-offering approach. A new technique based on L-shaped distributions underpins the impossibility proofs, offering a robust method to bound the power of Bayesian mechanisms in correlated settings.

Abstract

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 $1+\frac {\ln 2} 2\approx 1.346$.

Figures (8)

• 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.
• Figure 2: All possible allocation functions of $\mathcal{F}$ that trade the item in at least two instances.
• Figure 3: All allocation functions of $\mathcal{F}$ that trade the item in at most one instance.
• 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 .
• Figure 5: An L-shaped distribution $\mathcal{F}_k$ and the allocation rule with the buyer's threshold $b_i$, the seller's threshold $s_j$, and no allocation in the instance $(0, b_k)$. $b_i$ is the threshold of the buyer for the instances in the yellow rectangle: $(0,b_1), \dots ,(0,b_{k-1})$. The item is only traded in the yellow rectangle when $b\geq b_i$. Furthermore, by Observation \ref{['k-k-threshold-obs']}, the price is the same for every $b_k> b\geq b_i$. $s_j$ is the threshold of the seller for the instances in the purple rectangle: $(s_2,b_k), \dots ,(s_k,b_k)$. The item is only traded in the purple rectangle, when $s\leq s_j$. Furthermore, by Observation \ref{['k-k-threshold-obs']}, the price is the same for every $s_1 < s\leq s_j$. Finally, as the item is not traded in the instance $(0,b_k)$, there is no * in the cell corresponding to this instance. 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 $?$.

Theorems & Definitions (51)

• Theorem 3.1
• Lemma 3.2
• proof
• Theorem 4.1
• Definition 4.2
• Claim 4.3
• Claim 4.4
• Claim 4.5
• Lemma 4.6
• proof : Proof of Lemma \ref{['lemma-seller-prefers-higher-prices']}