Best-of-Both-Worlds Fairness of the Envy-Cycle-Elimination Algorithm

TL;DR

This paper investigates the fairness guarantees of the Envy-Cycle-Elimination algorithm for dividing indivisible goods, demonstrating its potential and limitations in achieving ex-ante proportionality through randomization.

Contribution

It introduces a randomized variant of ECE that achieves ex-post EFX and ex-ante envy-freeness for two agents, and analyzes the failure of similar methods for three agents.

Findings

Randomized ECE computes ex-post EFX and ex-ante envy-freeness for two agents.

Natural randomizations of ECE fail to ensure ex-ante proportionality for three agents.

The algorithm's fairness guarantees are limited to specific agent counts.

Abstract

We consider the problem of fairly dividing indivisible goods among agents with additive valuations. It is known that an Epistemic EFX and $2/3$-MMS allocation can be obtained using the Envy-Cycle-Elimination (ECE) algorithm. In this work, we explore whether this algorithm can be randomized to also ensure ex-ante proportionality. For two agents, we show that a randomized variant of ECE can compute an ex-post EFX and ex-ante envy-free allocation in near-linear time. However, for three agents, we show that several natural randomization methods for ECE fail to achieve ex-ante proportionality.