Single-Sample Bilateral Trade with a Broker

TL;DR

This paper studies single-sample bilateral trade with a broker, showing simple mechanisms can approximate optimal trade gains and welfare under informational constraints, contrasting with inapproximability results for strategic brokers.

Contribution

Introduces a model for single-sample bilateral trade with a broker, analyzing mechanisms that achieve constant-factor approximations to optimal outcomes under minimal data.

Findings

Mechanisms achieve constant-factor approximation to first-best GFT and SW.

Small losses in approximation due to broker presence compared to no-broker setting.

Contrasts with inapproximability results for strategic brokers with full knowledge.

Abstract

We initiate the study of single-sample bilateral trade with a broker, drawing an analogy to the setting of single-sample bilateral trade without a broker considered in Babaioff et al. (2020) and Cai and Wu (2023). Our model captures the three-sided interaction in which a broker mediates trade between a buyer and seller, each described by a valuation distribution from which a single sample can be drawn. We consider two settings in particular: one where the valuation distributions of the buyer and seller are identical and one where the valuation distributions are stochastically ordered. We analyze simple mechanisms that rely only on a single sample from each agent's distribution and show that these mechanisms achieve constant-factor approximations to the first-best gains-from-trade (GFT), first-best social welfare (SW), and optimal profit under the standard monotone-hazard-rate assumption. We then complement these results with matching or nearly matching upper bounds on the GFT and SW of our mechanisms. Notably, in both settings, we observe fairly small losses in the approximation factors to the first-best GFT and first-best SW due to the existence of the broker (benchmarked against the corresponding approximation factors in the setting without a broker). Furthermore, our results stand in stark contrast to those of Hajiaghayi et al. (2025), who show inapproximability results under a strategic broker with full distributional knowledge. Our results provide insight into the design of data-efficient brokerage mechanisms for online marketplaces and decentralized trading platforms, where intermediaries must facilitate trade under severe informational constraints. They highlight how even minimal data can enable robust and incentive-compatible brokerage in uncertain markets for both the broker and the market participants.