The Cost of EFX: Generalized-Mean Welfare and Complexity Dichotomies with Few Surplus Items

2026-01-19

AI Summary

This work investigates the allocation of indivisible goods when the number of agents is close to that of items—specifically, with at most three unallocated items—and explores how to optimize generalized p-mean welfare under the fairness criterion of EFX (envy-freeness up to any good). By integrating computational complexity theory, combinatorial optimization, and parameterized algorithms, the study establishes a sharp complexity dichotomy at \( p = 0 \): for \( p \leq 0 \), an EFX allocation maximizing welfare can be found in polynomial time, whereas for \( p > 0 \), the problem becomes NP-hard, with welfare loss growing linearly in the number of agents. Moreover, the paper proves that jointly verifying EFX and Pareto optimality is NP-hard, and that a stronger variant of EFX is \( \Sigma_2^P \)-complete, thereby revealing a fundamental trade-off between fairness and efficiency.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Search and Optimization: Combinatorial Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Envy-freeness up to any good (EFX) is a central fairness notion for allocating indivisible goods, yet its existence is unresolved in general. In the setting with few surplus items, where the number of goods exceeds the number of agents by a small constant (at most three), EFX allocations are guaranteed to exist, shifting the focus from existence to efficiency and computation. We study how EFX interacts with generalized-mean ($p$-mean) welfare, which subsumes commonly-studied utilitarian ($p=1$), Nash ($p=0$), and egalitarian ($p \rightarrow -\infty$) objectives. We establish sharp complexity dichotomies at $p=0$: for any fixed $p \in (0,1]$, both deciding whether EFX can attain the global $p$-mean optimum and computing an EFX allocation maximizing $p$-mean welfare are NP-hard, even with at most three surplus goods; in contrast, for any fixed $p \leq 0$, we give polynomial-time algorithms that optimize $p$-mean welfare within the space of EFX allocations and efficiently certify when EFX attains the global optimum. We further quantify the welfare loss of enforcing EFX via the price of fairness framework, showing that for $p>0$, the loss can grow linearly with the number of agents, whereas for $p \leq 0$, it is bounded by a constant depending on the surplus (and for Nash welfare it vanishes asymptotically). Finally we show that requiring Pareto-optimality alongside EFX is NP-hard (and becomes $\Sigma_2^P$-complete for a stronger variant of EFX). Overall, our results delineate when EFX is computationally costly versus structurally aligned with welfare maximization in the setting with few surplus items.

Problem

• EFX
• generalized-mean welfare
• fair division
• computational complexity
• price of fairness

Innovation

• EFX
• generalized-mean welfare
• complexity dichotomy
• price of fairness
• Pareto-optimality