Cesa-Bianchi, Nicolo, Tommaso, Cesari, Colomboni, Roberto, Fusco, Federico and Leonardi, Stefano (2021) A regret analysis of bilateral trade. EC '21: Proceedings of the 22nd ACM Conference on Economics and Computation. pp. 289-309.

Abstract

Bilateral trade, a fundamental topic in economics, models the problem of intermediating between two strategic agents, a seller and a buyer, willing to trade a good for which they hold private valuations. Despite the simplicity of this problem, a classical result by Myerson and Satterthwaite (1983) affirms the impossibility of designing a mechanism which is simultaneously efficient, incentive compatible, individually rational, and budget balanced. This impossibility result fostered an intense investigation of meaningful trade-offs between these desired properties. Much work has focused on approximately efficient fixed-price mechanisms, i.e., Blumrosen and Dobzinski (2014; 2016), Colini-Baldeschi et al. (2016), which have been shown to fully characterize strong budget balanced and ex-post individually rational direct revelation mechanisms. All these results, however, either assume some knowledge on the priors of the seller/buyer valuations, or a black box access to some samples of the distributions, as in D{ü}tting et al. (2021). In this paper, we cast for the first time the bilateral trade problem in a regret minimization framework over rounds of seller/buyer interactions, with no prior knowledge on the private seller/buyer valuations. Our main contribution is a complete characterization of the regret regimes for fixed-price mechanisms with different models of feedback and private valuations, using as benchmark the best fixed price in hindsight. More precisely, we prove the following bounds on the regret:
∙ Θ˜(T−−√) for full-feedback (i.e., direct revelation mechanisms);
∙ Θ˜(T2/3) for realistic feedback (i.e., posted-price mechanisms) and independent seller/buyer valuations with bounded densities;
∙ Θ(T) for realistic feedback and seller/buyer valuations with bounded densities;
∙ Θ(T) for realistic feedback and independent seller/buyer valuations;
∙ Θ(T) for the adversarial setting.