(Almost Full) EFX Exists for Four Agents (and Beyond)

Abstract

The existence of EFX allocations is a major open problem in fair division, even for additive valuations. The current state of the art is that no setting where EFX allocations are impossible is known, and EFX is known to exist for (i) agents with identical valuations, (ii) 2 agents, (iii) 3 agents with additive valuations, (iv) agents with one of two additive valuations and (v) agents with two valued instances. It is also known that EFX exists if one can leave n − 1 items unallocated, where n is the number of agents.We develop new techniques that allow us to push the boundaries of the enigmatic EFX problem beyond these known results, and, arguably, to simplify proofs of earlier results. Our main results are (i) every setting with 4 additive agents admits an EFX allocation that leaves at most a single item unallocated, (ii) every setting with n additive valuations has an EFX allocation with at most n − 2 unallocated items.Moreover, all of our results extend beyond additive valuations to all nice cancelable valuations (a new class, including additive, unit-demand, budget-additive and multiplicative valuations, among others). Furthermore, using our new techniques, we show that previous results for additive valuations extend to nice cancelable valuations.

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

“…While the existence of EFX allocations is open, it is known that there always exists an EF1 allocations for any number of agents, and it can be computed in polynomial time [29]. There are a lot of studies on EF1 and EFX [1,14,7,31,9,13,18,16,17,8,30]. Another major concept of fairness is maximin share (MMS), which was introduced by Budish [11].…”

“…Lemma 8. (Berger et al [8]) Let X be an allocation. If M X contains a Pareto improvable cycle, then there exists an allocation Y Pareto dominating X such that for any i, j ∈ N , if i does not EF X envy j in X, then neither in Y .…”

“…Note that if Y Pareto dominates X then Y lex X, but not vice versa. The following basic lemma is shown in [8], which we also use. Lemma 11.…”

“…Lemma 11. (Berger et al [8]) If for every partial EFX allocation X with k unallocated items, there exists an EFX allocation Y such that Y lex X, then there exists an EFX allocation with at most k − 1 unallocated items. Moreover, no agent envies the set of k − 1 unallocated items.…”

“…For additive valuations, Caragiannis et al [13] showed that there exists a partial EFX allocation where every agent receives at least half the value of her bundle in an optimal Nash social welfare allocation 2 . Quite recently, Berger et al [8] showed that when n = 4, there exists an EFX allocation with at most one unallocated item such that no agent envies the unallocated item. Moreover, they extend their results and existing results in [16] and [30] beyond additive valuations to nice cancelable valuations which is a class including additive, unit-demand, budget-additive, multiplicative valuations, and so on.…”