Optimizing the Envy Cycle Elimination Algorithm

2026-06-01

AI Summary

This work addresses the central challenge in fair allocation of indivisible goods: simultaneously achieving EF1 (envy-freeness up to one good) fairness and maximizing social welfare. The authors propose a heuristic strategy that jointly optimizes both item assignment and recipient selection, integrated within the envy-cycle elimination framework, thereby overcoming the limitations of conventional approaches that optimize only a single dimension. Theoretical analysis demonstrates that this method substantially improves the lower bound on utilitarian welfare. Empirical evaluations further confirm its effectiveness in reducing welfare loss on average, successfully balancing fairness and efficiency in practical settings.

Abstract

In the fair allocation of indivisible goods, a widely used notion of fairness is envy-freeness up to one good (EF1). A classical way to compute an EF1 allocation is the envy cycle elimination (ECE) algorithm, which iteratively assigns a good to an unenvied agent and, after each assignment, resolves any resulting envy cycle. Although the ECE algorithm always produces an EF1 allocation, it leaves considerable freedom in choosing both the next good to allocate and the agent to receive it. We investigate natural heuristics that exploit this flexibility to improve welfare guarantees. For example, we show that if the heuristic jointly selects the good and the receiving agent maximizing the utility, the worst-case utilitarian welfare loss is significantly lower than that of the vanilla algorithm. By contrast, restricting the heuristic to select only one of these two dimensions does not yield comparable improvements. We also complement our theoretical results with empirical average-case analysis.

Problem

• fair allocation
• EF1
• envy cycle elimination
• indivisible goods
• welfare guarantees

Innovation

• envy cycle elimination
• EF1 fairness
• welfare optimization
• indivisible goods allocation
• heuristic design