Gains-from-Trade in Bilateral Trade with a Broker

TL;DR

This paper analyzes gains-from-trade in bilateral trade when a strategic broker mediates between buyer and seller. It establishes both positive and negative results: for posted-pricing, constant-factor guarantees to the first-best $GFT$ hold under symmetric $MHR$ and under hazard-rate-bounded regimes, with a $1/36$ and a $2/M^2$-type bound respectively, and a quantile-based pricing approach yields a $\beta(1-\alpha)$-fraction of $FB-GFT$ under specific quantile relations. Beyond posted-pricing, the authors show a tight $1/2$-approximation for uniform distributions when the broker’s mechanism maximizes profit, but prove that for general distributions the approximation can be arbitrarily bad, including in public-agent and symmetric settings, and that similar inapproximability results extend to social welfare. The results illuminate when broker-mediated trade can closely approach the ideal social-welfare maximum and when strategic arbitrage by the broker can severely degrade gains-from-trade, guiding regulatory and mechanism-design considerations in brokered markets. Overall, the paper highlights a nuanced boundary between feasible constant-factor approximations and inherent limitations introduced by broker strategic behavior, dependent on distributional assumptions and the mechanism class employed.

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.

Theorems & Definitions (35)

• Theorem 1.1: Decomposition
• Corollary 1.2
• Theorem 1.3: Symmetric agents, MHR distributions
• Theorem 1.4: Symmetric agents, MHR distributions with bounded hazard rates
• Theorem 1.5: Quantile-based posted-pricing mechanism
• Corollary 1.6
• Theorem 1.7: myerson1983efficient
• Theorem 1.8: Uniform distribution
• Theorem 1.9: Inapproximability
• Theorem 1.10: Inapproximability with public seller or public buyer