Temporal Fair Division of Indivisible Items
Edith Elkind, Alexander Lam, Mohamad Latifian, Tzeh Yuan Neoh, Nicholas Teh
TL;DR
It is proved that determining the existence of a TEF1 allocation is NP-hard, and that TEF1 is incompatible with Pareto optimality, with the implication that it is intractable to find a TEF1 allocation that maximizes any p-mean welfare, even for two agents.
Abstract
We study a fair division model where indivisible items arrive sequentially, and must be allocated immediately and irrevocably. Previous work on online fair division has shown impossibility results in achieving approximate envy-freeness under these constraints. In contrast, we consider an informed setting where the algorithm has complete knowledge of future items, and aim to ensure that the cumulative allocation at each round satisfies approximate envy-freeness -- which we define as temporal envy-freeness up to one item (TEF1). We focus on settings where items can be exclusively goods or exclusively chores. For goods, while TEF1 allocations may not always exist, we identify several special cases where they do -- two agents, two item types, generalized binary valuations, unimodal preferences -- and provide polynomial-time algorithms for these cases. We also prove that determining the existence of a TEF1 allocation is NP-hard. For chores, we establish analogous results for the special cases, but present a slightly weaker intractability result. We also establish the incompatibility between TEF1 and Pareto-optimality, with the implication that it is intractable to find a TEF1 allocation that maximizes any $p$-mean welfare, even for two agents.