Gains-from-Trade in Bilateral Trade with a Broker

2024-10-22

arXiv.org

AI Summary

This paper studies the efficiency loss—measured by the gain from trade (GFT)—in bilateral trade mediated exclusively by a profit-maximizing broker, relative to the first-best (FB) benchmark. Using mechanism design and game-theoretic modeling under incentive compatibility (IC) and individual rationality (IR) constraints, we derive the first tight approximation ratios for GFT. Specifically, when buyer and seller valuations are uniformly distributed, the optimal GFT achieves exactly half the FB value—establishing a tight 1/2 approximation ratio. Under symmetric monotone hazard rate (MHR) distributions, we prove a lower bound of 1/36. Crucially, we construct counterexamples showing that, for arbitrary valuation distributions—including degenerate or symmetric ones—the GFT approximation ratio can deteriorate arbitrarily, implying no universal constant lower bound exists. These results reveal the fundamental efficiency cost imposed by intermediary arbitrage and provide a theoretical benchmark for evaluating intermediaries’ roles in platform economies.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Decision/Utility Theory

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• Economic Benefits
• Broker-mediated Transactions
• Bilateral Trade

Innovation

• Broker-Assisted Trading Model
• Revenue Lower Bound Quantification
• Optimization Infeasibility under Fairness