Allocating Indivisible Goods to Strategic Agents: Pure Nash Equilibria and Fairness

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 se‹ing 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), andwe positively answer the above question. In particular, we study two algorithms that are known to produce such allocations in the non-strategic se‹ing: Round-Robin (EF1 allocations for any number of agents) and a cut-and-choose algorithm of Plaut and Roughgarden [42] (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, while 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 non-strategic se‹ing! Further, we show that a weaker version of the la‹er result holds for any mechanism for two agents that always has pure Nash equilibria which all induce EFX allocations.