Round-Robin Beyond Additive Agents: Existence and Fairness of Approximate Equilibria

Fair allocation of indivisible goods has attracted extensive attention over the last two decades, yielding numerous elegant algorithmic results and producing challenging open questions. The problem becomes much harder in the presence of strategic agents. Ideally, one would want to design truthful mechanisms that produce allocations with fairness guarantees. However, in the standard setting without monetary transfers, it is generally impossible to have truthful mechanisms that provide nontrivial fairness guarantees. Recently, Amanatidis et al. [Amanatidis G, Birmpas G, Fusco F, Lazos P, Leonardi S, Reiffenhäuser R (2023) Allocating indivisible goods to strategic agents: Pure Nash equilibria and fairness. Math. Oper. Res. , ePub ahead of print November 30, https://doi.org/10.1287/moor.2022.0058 ] suggested the study of mechanisms that produce fair allocations in their equilibria. Specifically, when the agents have additive valuation functions, the simple Round-Robin algorithm always has pure Nash equilibria, and the corresponding allocations are envy-free up to one good (EF1) with respect to the agents’ true valuation functions . Following this agenda, we show that this outstanding property of the Round-Robin mechanism extends much beyond the above default assumption of additivity. In particular, we prove that for agents with cancelable valuation functions (a natural class that contains, e.g., additive and budget-additive functions), this simple mechanism always has equilibria, and even its approximate equilibria correspond to approximately EF1 allocations with respect to the agents’ true valuation functions. Furthermore, we show that the approximate EF1 fairness of approximate equilibria surprisingly holds for the important class of submodular valuation functions as well, even though exact equilibria fail to exist. Funding: This work was supported by the Horizon 2020 European Research Council Advanced Grant “AMDROMA: Algorithmic and Mechanism Design Research in Online Markets” [Grant 788893], the Ministero dell’Università e della Ricerca Research project of national interest (PRIN) “ALGADIMAR: Algorithms, Games, and Digital Markets,” the Nederlandse Organisatie voor Wetenschappelijk Onderzoek Veni Project “Algorithmic Fair Division in Dynamic, Socially Constrained Environments” [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].