Profit Maximization in Bilateral Trade against a Smooth Adversary

TL;DR

This paper introduces a learning algorithm for bilateral trade that achieves near-optimal regret bounds against a smooth adversary, extending fast convergence guarantees to more complex economic settings.

Contribution

It develops a novel algorithm with tight regret bounds for profit maximization in bilateral trade against smooth adversaries, broadening applicability of fast learning rates.

Findings

Achieves $ ilde{O}( oot{T} ull)$ regret bound in bilateral trade

Extends strong regret guarantees from i.i.d. to smooth adversaries

Applies techniques to the joint ads problem with similar bounds

Abstract

Bilateral trade models the task of intermediating between two strategic agents, a seller and a buyer, who wish to trade a good. We study this problem from the perspective of a profit-maximizing broker within an online learning framework, where the agents' valuations are generated by a smooth adversary. We devise a learning algorithm that guarantees a $\tilde{O}(\sqrt{T})$ regret bound, which is tight in the time horizon $T$ up to poly-logarithmic factors. This matches the minimax rate for the stochastic i.i.d. case, and is also well separated from the adversarial setting, where sublinear-regret is unattainable. By extending the strong regret guarantees from the i.i.d. case to the smooth adversary, we significantly broaden the scope of settings where such fast rate is achievable, while closing an important gap in the regret landscape of this fundamental economic problem. To overcome the challenges posed by this adversary, we leverage a continuity property of smooth instances and combines this with a hierarchical net-construction of the broker's action space, which is analyzed via algorithmic chaining. We showcase the applicability of these techniques by deriving a similarly tight $\tilde{O}(\sqrt{T})$ regret bound for a related mechanism design model: the joint ads problem.