Fixed-Price Approximations in Bilateral Trade

TL;DR

This paper studies fixed-price mechanisms in bilateral trade, improving approximation ratios for welfare maximization under various distribution knowledge assumptions, including identical distributions and limited samples.

Contribution

It provides new approximation bounds for welfare maximization in bilateral trade, including optimal ratios for identical distributions and improvements with limited prior samples.

Findings

Optimal welfare approximation ratio is (2+√2)/4 for identical distributions.

A 3/4-approximation to welfare is achievable with one prior sample.

The best-known (1-1/e)-approximation can be improved when distributions differ, but is optimal with only seller's distribution known.

Abstract

We consider the bilateral trade problem, in which two agents trade a single indivisible item. It is known that the only dominant-strategy truthful mechanism is the fixed-price mechanism: given commonly known distributions of the buyer's value $B$ and the seller's value $S$, a price $p$ is offered to both agents and trade occurs if $S \leq p \leq B$. The objective is to maximize either expected welfare $\mathbb{E}[S + (B-S) \mathbf{1}_{S \leq p \leq B}]$ or expected gains from trade $\mathbb{E}[(B-S) \mathbf{1}_{S \leq p \leq B}]$. We improve the approximation ratios for several welfare maximization variants of this problem. When the agents' distributions are identical, we show that the optimal approximation ratio for welfare is $\frac{2+\sqrt{2}}{4}$. With just one prior sample from the common distribution, we show that a $3/4$-approximation to welfare is achievable. When agents' distributions are not required to be identical, we show that a previously best-known $(1-1/e)$-approximation can be strictly improved, but $1-1/e$ is optimal if only the seller's distribution is known.