We consider the fundamental problem of allocating a set of indivisible goods among strategic agents with additive valuation functions. It is well known that, in the absence of monetary transfers, Pareto efficient and truthful rules are dictatorial, while there is no deterministic truthful mechanism that allocates all items and achieves envy-freeness up to one item (EF1), even for the case of two agents. In this paper, we investigate the interplay of fairness and efficiency under a relaxation of truthfulness called non-obvious manipulability (NOM), recently proposed by Troyan and Morrill. We show that this relaxation allows us to bypass the aforementioned negative results in a very strong sense. Specifically, we prove that there are deterministic and EF1 algorithms that are not obviously manipulable, and the algorithm that maximizes utilitarian social welfare (the sum of agents' utilities), which is Pareto efficient but not dictatorial, is not obviously manipulable for $n \geq 3$ agents (but obviously manipulable for $n=2$ agents). At the same time, maximizing the egalitarian social welfare (the minimum of agents' utilities) or the Nash social welfare (the product of agents' utilities) is obviously manipulable for any number of agents and items. Our main result is an approximation preserving black-box reduction from the problem of designing EF1 and NOM mechanisms to the problem of designing EF1 algorithms. En route, we prove an interesting structural result about EF1 allocations, as well as new "best-of-both-worlds" results (for the problem without incentives), that might be of independent interest.

翻译：我们研究在具有可加估值函数的策略性主体之间分配一组不可分割物品的基本问题。众所周知，在没有货币转移的情况下，帕累托有效且诚实的规则是独裁式的，而没有任何确定性诚实机制能够在分配所有物品的同时实现至多一件物品的无嫉妒性（EF1），即使对于两个主体的情形也是如此。本文在特罗扬和莫里尔最近提出的非明显可操纵性（NOM）这一诚实性松弛概念下，探讨公平性与效率之间的相互作用。我们证明，这种松弛使我们能够以非常强的意义规避上述负面结论。具体而言，我们证明存在确定性的、满足EF1且非明显可操纵的算法，而最大化功利主义社会福利（主体效用之和）的算法——该算法帕累托有效但不具独裁性——对于$n \geq 3$个主体是非明显可操纵的（但对于$n=2$个主体是明显可操纵的）。同时，最大化平等主义社会福利（主体效用的最小值）或纳什社会福利（主体效用的乘积）对于任意数量的主体和物品都是明显可操纵的。我们的主要结果是从设计EF1且NOM机制的问题到设计EF1算法的问题的一个保近似黑箱归约。在此过程中，我们还证明了关于EF1分配的一个有趣结构结果，以及可能具有独立意义的新的“两全其美”结果（针对无激励问题）。