Revisiting Fair and Efficient Allocations for Bivalued Goods

2026-04-09

AI Summary

This work addresses the inherent tension between fairness and efficiency in the allocation of indivisible goods under additive binary valuations, where existing algorithms often fail to terminate. The authors propose a polynomial-time algorithm that integrates combinatorial optimization and graph-theoretic techniques with item reallocations and weight adjustments to simultaneously achieve full Pareto optimality (fPO) and either weighted envy-freeness up to any good (WEFX) or weighted equitability up to any good (WEQX). This approach not only resolves the non-termination issues present in prior methods but also extends the theoretical frontier of fair division for binary-valued items, offering a novel paradigm for multi-agent resource allocation that combines computational efficiency with strong formal guarantees.

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.

Problem

• fair allocation
• indivisible goods
• bivalued valuations
• EFX
• fPO

Innovation

• WEFX
• fPO
• bivalued valuations
• polynomial-time algorithm
• fair allocation