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 above question. In particular, we study two algorithms that are known to produce such allocations in the non-strategic setting: Round-Robin (EF1 allocations for any number of agents) and a cut-and-choose algorithm of Plaut and Roughgarden [SIAM Journal of Discrete Mathematics, 2020] (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 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.

翻译：我们考虑将一组不可分割物品公平分配给一组具有加性估值函数的策略性代理人的问题。我们假设不涉及货币转移，因此在此设置中，机制是一个以代理人报告值（而非真实值）作为输入的算法。我们的主要目标是探索是否存在针对每个实例都具有纯纳什均衡的机制，同时为这些均衡对应的分配提供公平性保证。我们专注于无嫉妒性的两种松弛形式，即最多一种物品的无嫉妒性（EF1）和任意一种物品的无嫉妒性（EFX），并对上述问题给出肯定回答。具体而言，我们研究了两种在非策略性设置中已知能产生此类分配的算法：循环法（针对任意数量代理人产生EF1分配）以及Plaut和Roughgarden提出的剪切与选择算法[SIAM离散数学期刊，2020]（针对两个代理人产生EFX分配）。对于循环法，我们证明其所有纯纳什均衡都会诱导出基于真实值的EF1分配；而对于Plaut和Roughgarden的算法，我们证明相应的分配不仅具有EFX性质，还满足最大最小份额公平性，而这一点在该算法的非策略性设置中并不成立！此外，我们证明后一结果的弱化版本适用于任何针对两个代理人且始终存在纯纳什均衡并诱导EFX分配的机制。