Stefano Leonardi

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. In this paper, we cast the bilateral trade problem in a regret minimization framework over T rounds of seller/buyer interactions, with no prior knowledge on their private valuations. Our main contribution is a complete characterization of the regret regimes for fixed-price mechanisms with different feedback models and private valuations, using as a benchmark the best fixed price in hindsight. More precisely, we prove the following tight bounds on the regret: [Formula: see text] for full-feedback (i.e., direct revelation mechanisms). [Formula: see text] for realistic feedback (i.e., posted-price mechanisms) and independent seller/buyer valuations with bounded densities. [Formula: see text] for realistic feedback and seller/buyer valuations with bounded densities. [Formula: see text] for realistic feedback and independent seller/buyer valuations. [Formula: see text] for the adversarial setting. Funding: This work was partially supported by the European Research Council Advanced [Grant 788893] AMDROMA “Algorithmic and Mechanism Design Research in Online Markets”, the Ministero dell’Istruzione, dell’Università e della Ricerca PRIN project ALGADIMAR “Algorithms, Games, and Digital Markets”, the AI Interdisciplinary Institute ANITI (funded by the French “Investing for the Future—PIA3” program under the [Grant agreement ANR-19-PI3A-0004], the project BOLD from the French national research agency (ANR), the EU Horizon 2020 ICT-48 research and innovation action ELISE (European Learning and Intelligent Systems Excellence, [Grant agreement 951847].

We consider the problem of fairly allocating a set of indivisible goods to a set of strategic agents with additive valuation functions. We assume no monetary transfers, and therefore, a mechanism in our setting is an algorithm that takes as input the reported—rather than the true—values of the agents. Our main goal is to explore whether there exist mechanisms that have pure Nash equilibria for every instance and, at the same time, provide fairness guarantees for the allocations that correspond to these equilibria. We focus on two relaxations of envy-freeness, namely, envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX), and we positively answer the preceding question. In particular, we study two algorithms that are known to produce such allocations in the nonstrategic setting: round-robin (EF1 allocations for any number of agents) and a cut-and-choose algorithm of Plaut and Roughgarden (EFX allocations for two agents). For round-robin, we show that all of its pure Nash equilibria induce allocations that are EF1 with respect to the underlying true values, whereas for the algorithm of Plaut and Roughgarden, we show that the corresponding allocations not only are EFX, but also satisfy maximin share fairness, something that is not true for this algorithm in the nonstrategic setting! Further, we show that a weaker version of the latter result holds for any mechanism for two agents that always has pure Nash equilibria, which all induce EFX allocations. Funding: This work was supported by the Horizon 2020 European Research Council Advanced “Algorithmic and Mechanism Design Research in Online Markets” [Grant 788893], the Ministero dell’Università e della Ricerca Research project of national interest (PRIN) “Algorithms, Games, and Digital Markets,” the Future Artificial Intelligence Research project funded by the NextGenerationEU program within the National Recovery and Resilience Plan (PNRR-PE-AI) scheme [M4C2, investment 1.3, line on Artificial Intelligence], the National Recovery and Resilience Plan-Ministero dell’Università e della Ricerca (PNRR-MUR) project IR0000013-SoBigData.it, the Nederlandse Organisatie voor Wetenschappelijk Onderzoek Veni Project [Grant VI.Veni.192.153], and the National Recovery and Resilience Plan Greece 2.0 funded by the European Union under the NextGenerationEU Program [Grant MIS 5154714].