Multiple Birds with One Stone: Beating 1/2 for EFX and GMMS via Envy Cycle Elimination

Abstract

Several relaxations of envy-freeness, tailored to fair division in settings with indivisible goods, have been introduced within the last decade. Due to the lack of general existence results for most of these concepts, great attention has been paid to establishing approximation guarantees. In this work, we propose a simple algorithm that is universally fair in the sense that it returns allocations that have good approximation guarantees with respect to four such fairness notions at once. In particular, this is the first algorithm achieving a (φ−1)-approximation of envy-freeness up to any good (EFX) and a 2/φ+2 -approximation of groupwise maximin share fairness (GMMS), where φ is the golden ratio. The best known approximation factor, in polynomial time, for either one of these fairness notions prior to this work was 1/2. Moreover, the returned allocation achieves envy-freeness up to one good (EF1) and a 2/3-approximation of pairwise maximin share fairness (PMMS). While EFX is our primary focus, we also exhibit how to fine-tune our algorithm and improve further the guarantees for GMMS or PMMS.Finally, we show that GMMS—and thus PMMS and EFX—allocations always exist when the number of goods does not exceed the number of agents by more than two.

Optimal Bounds on the Price of Fairness for Indivisible Goods

“…While this preserves exact envy-freeness in the cake-cutting setting of Arunachaleswaran et al [3], in our setting we argue that inclusion-wise minimality preserves the EF1 property of P. We also argue that this iterative process terminates at a partial allocation P that satisfies the desired social welfare guarantee. Finally, we use the algorithm of Lipton et al [20] (see also [2]) to extend this partial EF1 allocation into a complete EF1 allocation without losing social welfare. The detailed algorithm is presented as Algorithm 2.…”

“…This generalizes the classical cut-and-choose protocol for dividing a cake between two agents, where one agent cuts the cake into two pieces, the other agent chooses a piece, and the first agent then receives the remaining piece. Even for agents with additive valuations over the goods, 2 MMS allocations may not exist [18,23]. However, under additive valuations, an allocation where each agent receives at least 3 /4-th of her maximin share-i.e., a 3 /4-MMS allocation-always exists [16,17].…”

Equitable Division of a Path

“…4 Similar approximations have been studied in the context of envy-freeness up to any good (EFX) (Plaut and Roughgarden, 2020;Amanatidis et al, 2020).…”

Extension of Additive Valuations to General Valuations on the Existence of EFX

“…We also study the case with a small number of items. For additive valuations, Amanatidis et al [3] showed that when m ≤ n + 2, there exists an EFX allocation. For general valuations, to the best of our knowledge, non-trivial results are not known.…”