Bilateral Trade with Correlated Values

TL;DR

This paper investigates bilateral trade mechanisms with correlated buyer and seller values, establishing optimal approximation ratios for social welfare and highlighting limitations of incentive-compatible mechanisms.

Contribution

It introduces a buyer-offering mechanism that achieves near-optimal social welfare approximation under correlated values and proves fundamental limits of mechanism performance.

Findings

Buyer-offering mechanism guarantees 1.582 approximation ratio.

No mechanism with both sides having dominant strategies can achieve constant approximation.

Impossibility results for deterministic Bayesian incentive-compatible mechanisms.

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