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

TL;DR

This paper introduces a new algorithm for online bilateral trade that achieves sublinear regret against the global budget balance benchmark, improving social welfare in stochastic environments with one-bit feedback.

Contribution

It presents the first algorithm to attain sublinear regret against the GBB benchmark in stochastic settings with limited feedback, bridging the gap between WBB and GBB constraints.

Findings

Achieves rac{3}{4} regret in stochastic environments.

Demonstrates no separation between learning WBB prices and GBB distributions.

Provides a new benchmark for online bilateral trade algorithms.

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.