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, we can always find (i) a sequence of allocations that is envy-free and complete (in polynomial time), and (ii) a sequence of allocations that is proportional and Pareto-optimal (in exponential time). On the other hand, we show that 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.

翻译：公平分配一组不可分物品是（计算）社会选择领域中的一个著名难题。在这种情况下，公平概念（如无嫉妒性和比例性）与经济效率（如帕累托最优性）之间存在根本性矛盾。然而，在现实世界中，物品并非总是被一次性分配，而是经常重复分配。例如，这些物品可能是家庭中需要重复分配的日常杂务。受此启发，我们首次研究了不可分物品与杂务的重复公平分配问题，并为此场景提出了一个形式化模型。本文证明，若重复次数是代理人数的整数倍，则总能（i）在多项式时间内找到一个无嫉妒且完整的分配序列，以及（ii）在指数时间内找到一个比例且帕累托最优的分配序列。另一方面，我们证明无论重复次数如何，可能不存在无嫉妒且帕累托最优的分配序列。对于两个代理人的情况，我们证明若重复次数为偶数，则总能找到一个整体无嫉妒且帕累托最优的分配序列。我们进一步证明了更强的公平性保证，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。