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 non-trivial fairness guarantees. Recently, Amanatidis et al. [2021] 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. Further, 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!

翻译：不可分割物品的公平分配在过去二十年中引起了广泛关注，产生了众多精巧的算法成果，并提出了具有挑战性的开放问题。在存在策略性智能体的情况下，该问题变得更为困难。理想情况下，我们希望设计能产生具有公平性保障的分配结果的真实机制。然而，在无货币转移的标准设定下，通常不可能设计出能提供非平凡公平性保障的真实机制。近期，Amanatidis 等人 [2021] 提出研究在均衡状态下能产生公平分配结果的机制。具体而言，当智能体具备可加性估值函数时，简单的轮询算法始终存在纯纳什均衡，且相应的分配结果相对于智能体的真实估值函数满足"至多一件物品的无嫉妒性"（EF1）。沿此研究方向，我们证明轮询机制的这一卓越性质远远超越了上述默认的可加性假设。特别地，对于具有可取消性估值函数（一类包含可加性函数和预算可加性函数等自然函数类）的智能体，我们证明这一简单机制始终存在均衡，且甚至其近似均衡也对应于相对于智能体真实估值函数的近似 EF1 分配。进一步地，我们惊人地发现，即使当精确均衡不存在时，针对子模估值函数这一重要函数类，近似均衡的近似 EF1 公平性仍然成立！