Envy-Freeness Up to Any Item with High Nash Welfare

Abstract

Several fairness concepts have been proposed recently in attempts to approximate envy-freeness in settings with indivisible goods. Among them, the concept of envy-freeness up to any item (EFX) is arguably the closest to envy-freeness. Unfortunately, EFX allocations are not known to exist except in a few special cases. We make significant progress in this direction. We show that for every instance with additive valuations, there is an EFX allocation of a subset of items with a Nash welfare that is at least half of the maximum possible Nash welfare for the original set of items. That is, after donating some items to a charity, one can distribute the remaining items in a fair way with high efficiency. This bound is proved to be best possible. Our proof is constructive and highlights the importance of maximum Nash welfare allocation. Starting with such an allocation, our algorithm decides which items to donate and redistributes the initial bundles to the agents, eventually obtaining an allocation with the claimed efficiency guarantee. The application of our algorithm to large markets, where the valuations of an agent for every item is relatively small, yields EFX with almost optimal Nash welfare. To the best of our knowledge, this is the first use of large market assumptions in the fair division literature. We also show that our algorithm can be modified to compute, in polynomial-time, EFX allocations that approximate optimal Nash welfare within a factor of at most 2ρ, using a ρ-approximate allocation on input instead of the maximum Nash welfare one.

EFX Exists for Three Agents

“…EFX with charity: Quite recently there have been studies [9,13] that consider relaxations of EFX, called "EFX with charity". Here we look for partial EFX allocations, where not all items need to be allocated (some of them remain unallocated).…”

“…for any agent i, we have v i (X i ) ≥ v i (P )), and P has less than n items (i.e., |P | < n), even in the case of general valuations. In case of additive valuations, Caragiannis et al [9] show the EC'20 Session 1a: New Solutions in Fair Division existence of a partial EFX allocation X = ⟨X 1 ,X 2 , . .…”

“…The Nash welfare of a fair allocation is often considered as a measure of its efficiency [9]: Intuitively, it captures how much average welfare the allocation achieves while still remaining fair. The result of Caragiannis et al [9] imply that there are efficient partial EFX allocations (partial EFX allocations with a 2-approximation of the maximum possible Nash welfare). Indeed, it is a natural question to ask whether there are complete EFX allocations (all items are allocated) with good efficiency.…”

“…Indeed, it is a natural question to ask whether there are complete EFX allocations (all items are allocated) with good efficiency. To this end, Caragiannis et al [9] conjecture: Conjecture 1. "In particular, we suspect that adding an item to an allocation problem (that provably has an EFX allocation) yields another problem that also has an EFX allocation with at least as high Nash welfare as the initial one."…”

“…This example also shows how EFX helps to rule out some unsatisfactory EF1 allocations. Caragiannis et al [9] remark that "Arguably, EFX is the best fairness analog of envy-freeness of indivisible items." While an EF1 allocation is always guaranteed to exist, very little is known about the existence of EFX allocations.…”