We study the classic problem of dividing a collection of indivisible resources in a fair and efficient manner among a set of agents having varied preferences. Pareto optimality is a standard notion of economic efficiency, which states that it should be impossible to find an allocation that improves some agent's utility without reducing any other's. On the other hand, a fundamental notion of fairness in resource allocation settings is that of envy-freeness, which renders an allocation to be fair if every agent (weakly) prefers her own bundle over that of any other agent's bundle. Unfortunately, an envy-free allocation may not exist if we wish to divide a collection of indivisible items. Introducing randomness is a typical way of circumventing the non-existence of solutions, and therefore, allocation lotteries, i.e., distributions over allocations have been explored while relaxing the notion of fairness to ex-ante envy freeness. We consider a general fair division setting with $n$ agents and a family of admissible $n$-partitions of an underlying set of items. Every agent is endowed with partition-based utilities, which specify her cardinal utility for each bundle of items in every admissible partition. In such fair division instances, Cole and Tao (2021) have proved that an ex-ante envy-free and Pareto-optimal allocation lottery is always guaranteed to exist. We strengthen their result while examining the computational complexity of the above total problem and establish its membership in the complexity class PPAD. Furthermore, for instances with a constant number of agents, we develop a polynomial-time algorithm to find an ex-ante envy-free and Pareto-optimal allocation lottery. On the negative side, we prove that maximizing social welfare over ex-ante envy-free and Pareto-optimal allocation lotteries is NP-hard.

翻译：我们研究在具有不同偏好的智能体间公平高效地分配不可分割资源集合的经典问题。帕累托最优性是经济效率的标准概念，它要求不存在能提升某个智能体效用同时不降低其他智能体效用的分配方案。而在资源分配场景中，无嫉妒性是基本公平概念——若每个智能体（弱）偏好自身所得组合而非其他智能体的组合，则该分配被视为公平。遗憾的是，对于不可分割物品的分配，无嫉妒分配可能不存在。引入随机性成为规避无解问题的典型方法，因此研究者通过将公平条件松弛为事前无嫉妒性，探索了分配随机机制（即分配的概率分布）。本文考虑包含n个智能体及基础物品集合的若干可容许n-划分的一般公平分配场景。每个智能体具备基于划分的效用函数——该函数量化其在每个可容许划分中各物品组合的基数效用。针对此类公平分配实例，Cole与Tao（2021）已证明必然存在满足事前无嫉妒性与帕累托最优性的分配随机机制。我们在考察上述总问题的计算复杂性时强化了其结论，证明了该问题属于PPAD复杂性类。进一步地，对于智能体数量为常数的实例，我们提出了在多项式时间内求解事前无嫉妒且帕累托最优分配随机机制的算法。在否定方面，我们证明在事前无嫉妒且帕累托最优的分配随机机制中最大化社会福利是NP难问题。