The problem of fairly allocating a set of indivisible items is a well-known challenge in the field of (computational) social choice. In this scenario, there is a fundamental incompatibility between notions of fairness (such as envy-freeness and proportionality) and economic efficiency (such as Pareto-optimality). However, in the real world, items are not always allocated once and for all, but often repeatedly. For example, the items may be recurring chores to distribute in a household. Motivated by this, we initiate the study of the repeated fair division of indivisible goods and chores, and propose a formal model for this scenario. In this paper, we show that, if the number of repetitions is a multiple of the number of agents, there always exists a sequence of allocations that is proportional and Pareto-optimal. On the other hand, irrespective of the number of repetitions, an envy-free and Pareto-optimal sequence of allocations may not exist. For the case of two agents, we show that if the number of repetitions is even, it is always possible to find a sequence of allocations that is overall envy-free and Pareto-optimal. We then prove even stronger fairness guarantees, showing that every allocation in such a sequence satisfies some relaxation of envy-freeness. Finally, in case that the number of repetitions can be chosen freely, we show that envy-free and Pareto-optimal allocations are achievable for any number of agents.

翻译：不可分割物品的公平分配问题是（计算）社会选择领域中一个众所周知的挑战。在此情境下，公平性概念（如无嫉妒性和比例性）与经济效率（如帕累托最优性）之间存在根本性的不相容。然而，在现实世界中，物品并非总是一次性分配完毕，而是经常重复分配。例如，家庭中可能反复出现待分配的家务杂事。受此启发，我们首次研究了不可分割商品与杂务的重复公平分配问题，并为此场景提出了一个正式模型。本文证明，若重复次数是代理人数的倍数，则始终存在一个既具备比例性又达到帕累托最优的分配序列。另一方面，无论重复次数多少，可能都不存在既无嫉妒又帕累托最优的分配序列。针对两个代理人的情形，我们证明：若重复次数为偶数，则总可能找到一个整体上无嫉妒且帕累托最优的分配序列。随后，我们进一步证明了更强的公平性保障，表明该序列中的每一次分配都满足某种无嫉妒性的松弛条件。最后，在重复次数可自由选择的情况下，我们证明对于任意数量的代理人，均可实现无嫉妒且帕累托最优的分配。