A Stronger Benchmark for Online Bilateral Trade: From Fixed Prices to Distributions

Abstract: We study online bilateral trade, where a learner facilitates repeated exchanges between a buyer and a seller to maximize the Gain From Trade (GFT), i.e., the social welfare. In doing so, the learner must guarantee not to subsidize the market. This constraint is usually imposed per round through Weak Budget Balance (WBB). Despite that, Bernasconi et al. [2024] show that a Global Budget Balance (GBB) constraint on the profit -- enforced over the entire time horizon -- can improve the GFT by a multiplicative factor of two. While this might appear to be a marginal relaxation, this implies that all existing WBB-focused algorithms suffer linear regret when measured against the GBB optimum. In this work, we provide the first algorithm to achieve sublinear regret against the GBB benchmark in stochastic environments under one-bit feedback. In particular, we show that when the joint distribution of valuations has a bounded density, our algorithm achieves $\widetilde{\mathcal{O}}(T^{3/4})$ regret. Our result shows that there is no separation between the one-dimensional problem of learning the optimal WBB price and the two-dimensional problem of learning the optimal GBB distribution over pairs of prices.