Sample Complexity of the Second-Best Bilateral Trade

Abstract

We study the sample complexity of learning near-optimal bilateral trade mechanisms. Unlike previous work on learning simple or fixed-price bilateral-trade mechanisms, we focus on mechanisms satisfying Bayesian incentive compatibility (BIC), interim individual rationality (IIR), and ex-ante weak budget balance (WBB). In other words, our target is to design a sample-based mechanism that achieves the second-best gains-from-trade benchmark. We give matching or nearly matching upper and lower bounds in three regimes. For regular product distributions on $[0,h]^2$, additive $\varepsilon$-approximation has sample complexity $\widetilde{\Theta}(h^2/\varepsilon^2)$. For multiplicative $(1-\alpha)$-approximation under the same assumptions, we find that the sample complexity is $\widetilde{\Theta}(h/(\mathrm{SB}(D)\alpha^2))$, which is benchmark-sensitive with unavoidable dependence on the second-best gains from trade $\mathrm{SB}(D)$. We also investigate unbounded distributions under a monotone hazard rate (MHR) assumption. The sample complexity depends on the ratio $\chi_\mu(D)=\mu(D)/\mathrm{SB}(D)$, where $\mu(D)$ is the sum of the buyer's expected value and the seller's expected cost.