Revisiting Fair and Efficient Allocations for Bivalued Goods

TL;DR

This paper revisits fair and efficient allocations of indivisible bivalued goods, correcting previous algorithms and proposing new methods for WEFX and WEQX allocations with proven polynomial-time complexity.

Contribution

It introduces a new polynomial-time algorithm for WEFX and fPO allocations, addressing failures in prior methods and extending to WEQX allocations.

Findings

Counterexample shows previous algorithm may fail to terminate.

New algorithm computes WEFX and fPO allocations in polynomial time.

Method can be adapted for WEQX and fPO allocations.

Abstract

This paper re-examines the problem of fairly and efficiently allocating indivisible goods among agents with additive bivalued valuations. Garg and Murhekar (2021) proposed a polynomial-time algorithm that purported to find an EFX and fPO allocation. However, we provide a counterexample demonstrating that their algorithm may fail to terminate. To address this issue, we propose a new polynomial-time algorithm that computes a WEFX (Weighted Envy-Free up to any good) and fPO allocation, thereby correcting the prior approach and offering a more general solution. Furthermore, we show that our algorithm can be adapted to compute a WEQX (Weighted Equitable up to any good) and fPO allocation.