On Optimal Tradeoffs between EFX and Nash Welfare

TL;DR

This work formalizes and tightens the trade-offs between fairness and efficiency in allocating indivisible goods by studying $\alpha$-EFX fairness alongside $\beta$-MNW efficiency. It provides constructive, polynomial-time algorithms that start from a maximum Nash welfare allocation to produce $\alpha$-EFX (and EF1) partial allocations with $\beta=\tfrac{1}{\alpha+1}$ MNW, and employs envy-cycles to extend these to complete allocations when $\alpha$ lies within key bounds: $\alpha \leq \varphi-1$ for additive valuations and $\alpha \leq \tfrac{1}{2}$ for subadditive valuations, achieving strong MNW guarantees (including $2/3$ MNW at $\alpha=\tfrac{1}{2}$). The results tighten previous bounds and reveal tight impossibility limits, illustrating the limits of simultaneous fairness and efficiency. Beyond α-EFX and MNW, the constructions also yield EF1 and MMS-type guarantees in the additive case, highlighting broader fairness benefits of the approach. Overall, the paper advances both theoretical understanding and practical procedures for fair and efficient resource allocation in settings with indivisible goods.

Abstract

A major problem in fair division is how to allocate a set of indivisible resources among agents fairly and efficiently. The goal of this work is to characterize the tradeoffs between two well-studied measures of fairness and efficiency -- envy freeness up to any item (EFX) for fairness, and Nash welfare for efficiency -- by saying, for given constants $α$ and $β$, whether there exists an $α$-EFX allocation that guarantees a $β$-fraction of the maximum Nash welfare ($β$-MNW). For additive valuations, we show that for any $α\in [0,1]$, there exists a partial allocation that is $α$-EFX and $\frac{1}{α+1}$-MNW, and this tradeoff is tight (for any $α$). We also show that for $α\in[0,\varphi-1 \approx 0.618]$ these partial allocations can be turned into complete allocations where all items are assigned. Furthermore, for any $α\in [0, 1/2]$, we show that the tight tradeoff of $α$-EFX and $\frac{1}{α+1}$-MNW with complete allocations holds for the more general setting of subadditive valuations. Our results improve upon the current state of the art, for both additive and subadditive valuations, and match the best-known approximations of EFX under complete allocations, regardless of Nash welfare guarantees. Notably, our constructions for additive valuations also provide EF1 and constant approximations for maximin share guarantees.

Figures (5)

• Figure 1: Trade-off between the existence of $\alpha$-EFX and $\beta$-MNW allocations, for additive (left) and subadditive (right) valuations. Previous positive results are represented by dots (complete dots for complete allocations, and hollow dots for partial allocations). Dots that lie on the curves belong to the regions of positive results. All positive results for additive valuations also guarantee EF1.
• Figure 2: The structure of the new allocation $\widehat{X}$ in Case 1 and Case 2. The edges represent the matching $M$.
• Figure 3: The structure of the new allocation $\widehat{X}$ in Case 4 and Case 5: the touching sequence has length $\ell >2$, and ends with condition (1). The edges represent the touching relation.
• Figure 4: Split operation, as defined in Algorithm \ref{['alg:split_bundle']}. The new allocation is described by the matching $M^t$. Items that are thrown away (not in any white, red, or blue bundle) are hatched. Regarding the new structure of $X_{j^t}$, in each case the bundle $Z_{j^t}^{t}$ is colored in white and the bundle $X_{j^t}^{t}\setminus Z_{j^t}^{t}$ is colored in red.
• Figure 5: The structure of the matching $M^t$ over the course of Algorithms \ref{['alg:new_subadditive_alg']}, as described in Claim \ref{['cla:chains']}. Each edge connects an agent $k$ to her bundle $M_k^t$. In the beginning, everyone is unmatched and there are only chains of type (i) of length $\ell=1$. In the end, everyone is matched and there are no chains of type (i).

Theorems & Definitions (90)

• Example 1
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4: Impossibility
• Definition 2.1: $\gamma$-separation
• Lemma 2.2
• Theorem 4
• Claim 3.0
• Lemma 3.1