An $\alpha$-regret analysis of Adversarial Bilateral Trade

TL;DR

This paper analyzes the limits of designing incentive-compatible mechanisms for adversarial bilateral trade, revealing fundamental bounds on regret and the impact of feedback and pricing strategies.

Contribution

It provides the first comprehensive regret analysis for adversarial bilateral trade, highlighting the separation between feedback scenarios and pricing strategies.

Findings

Sublinear 2-regret is achievable with full feedback.

Sublinear α-regret for α<2 is impossible in general.

Two-price posting with one-bit feedback achieves sublinear 2-regret.

Abstract

We study sequential bilateral trade where sellers and buyers valuations are completely arbitrary (i.e., determined by an adversary). Sellers and buyers are strategic agents with private valuations for the good and the goal is to design a mechanism that maximizes efficiency (or gain from trade) while being incentive compatible, individually rational and budget balanced. In this paper we consider gain from trade which is harder to approximate than social welfare. We consider a variety of feedback scenarios and distinguish the cases where the mechanism posts one price and when it can post different prices for buyer and seller. We show several surprising results about the separation between the different scenarios. In particular we show that (a) it is impossible to achieve sublinear $\alpha$-regret for any $\alpha<2$, (b) but with full feedback sublinear $2$-regret is achievable (c) with a single price and partial feedback one cannot get sublinear $\alpha$ regret for any constant $\alpha$ (d) nevertheless, posting two prices even with one-bit feedback achieves sublinear $2$-regret, and (e) there is a provable separation in the $2$-regret bounds between full and partial feedback.