Gains-from-Trade in Bilateral Trade with a Broker

TL;DR

This paper analyzes how a broker's strategic behavior affects the efficiency of bilateral trade, showing that certain mechanisms can guarantee a constant-factor approximation to optimal trade gains, but this can deteriorate in some cases.

Contribution

It provides bounds on the gains-from-trade approximation achievable with different mechanisms in bilateral trade with a broker, including tight bounds and limitations.

Findings

GFT achieves a 1/36-approximation with optimal posted-pricing mechanisms.

Arbitrary IC and IR mechanisms can guarantee a 1/2-approximation under uniform distributions.

The approximation can be arbitrarily poor in some instances, even with symmetric distributions.

Abstract

We study bilateral trade with a broker, where a buyer and seller interact exclusively through the broker. The broker strategically maximizes her payoff through arbitrage by trading with the buyer and seller at different prices. We study whether the presence of the broker interferes with the mechanism's gains-from-trade (GFT) achieving a constant-factor approximation to the first-best gains-from-trade (FB). We first show that the GFT achieves a $1 / 36$-approximation to the FB even if the broker runs an optimal posted-pricing mechanism under symmetric agents with monotone-hazard-rate distributions. Beyond posted-pricing mechanisms, even if the broker uses an arbitrary incentive-compatible (IC) and individually-rational (IR) mechanism that maximizes her expected profit, we prove that it induces a $1 / 2$-approximation to the first-best GFT when the buyer and seller's distributions are uniform distributions with arbitrary support. This bound is shown to be tight. We complement such results by proving that if the broker uses an arbitrary profit-maximizing IC and IR mechanism, there exists a family of problem instances under which the approximation factor to the first-best GFT becomes arbitrarily bad. We show that this phenomenon persists even if we restrict one of the buyer's or seller's distributions to have a singleton support, or even in the symmetric setting where the buyer and seller have identical distributions.