Almost Envy-Freeness with General Valuations

Abstract

The goal of fair division is to distribute resources among competing players in a "fair" way. Envy-freeness is the most extensively studied fairness notion in fair division. Envy-free allocations do not always exist with indivisible goods, motivating the study of relaxed versions of envy-freeness. We study the envy-freeness up to any good (EFX) property, which states that no player prefers the bundle of another player following the removal of any single good, and prove the first general results about this property. We use the leximin solution to show existence of EFX allocations in several contexts, sometimes in conjunction with Pareto optimality. For two players with valuations obeying a mild assumption, one of these results provides stronger guarantees than the currently deployed algorithm on Spliddit, a popular fair division website. Unfortunately, finding the leximin solution can require exponential time. We show that this is necessary by proving an exponential lower bound on the number of value queries needed to identify an EFX allocation, even for two players with identical valuations. We consider both additive and more general valuations, and our work suggests that there is a rich landscape of problems to explore in the fair division of indivisible goods with different classes of player valuations.removed; this is a thought experiment used in the definition of envy-freeness up to one good. An EF1 allocation always exists, and can be computed in polynomial time [20]. 1 Caragiannis et al. [10] proposed another fairness criterion, one which is strictly stronger than EF1, but strictly weaker than full envy-freeness. An allocation is envy-free up to any good (EFX) if for any i, j where player i envies player j, removing any good from j's allocation would eliminate i's envy. Do EFX allocations always exist? This paper takes the first steps toward answering this question.

Envy-Freeness Up to Any Item with High Nash Welfare

“…We employ the natural and practical operation of donating/removing items to resolve the challenging issues related to the EFX solution concept. Our approach contrasts with the recent attempts to mitigate such challenges by considering approximate versions of EFX (e.g., in the papers by Plaut and Roughgarden [33] and Amanatidis et al [1]). Instead, we keep the precise definition of EFX and use approximation only for the efficiency guarantee which is unavoidable.…”

“…Unlike EF1, the existence of EFX allocations is still a mystery, even for three agents with additive valuations. Plaut and Roughgarden [33] prove the existence of EFX allocations for setting with two agents only or with more agents and identical valuations. In addition, they present an algorithm for computing an 1/2-EFX allocation, where the value of each agent from her bundle is at least half of what the EFX property requires it to be.…”

When Do Envy-Free Allocations Exist?

“…These relaxations include envy-freeness up to one good-any envy that an agent has towards another agent can be eliminated by removing some item from the latter agent's bundle-and envy-freeness up to any goodany such envy can be eliminated by removing any item from the latter agent's bundle. It has been shown that these relaxations do provide existence guarantees in a number of settings (Lipton et al, 2004;Caragiannis et al, 2016;Biswas and Barman, 2018;Plaut and Roughgarden, 2018).…”

Maximin-Aware Allocations of Indivisible Goods

“…If the above requirement of submodularity is replaced by strictly increasing subadditivity, an MMAX allocation is guaranteed to exist. In contrast, MMS allocation may not exist even for three agents with additive valuations [Kurokawa et al, 2018] and an EFX allocation is only known to exist when there are two agents [Plaut and Roughgarden, 2018].…”

“…In [Caragiannis et al, 2016], the authors introduced a strictly stronger fairness notation than EF1 called envy-free up to any good (EFX), where the comparison is made to "any" single good instead of "a" single good. The state-of-the-art results show that an EFX allocation exists in the following settings: (1) there are 2 agents, or (2) there are any number of agents but all of them have the identical valuation [Plaut and Roughgarden, 2018]. It is still an open question whether an EFX allocation exists in general, even for additive valuations.…”