Probing EFX via PMMS: (Non-)Existence Results in Discrete Fair Division

2025-07-20

AI Summary

This paper investigates the existence of EFX (envy-freeness up to any good) and PMMS (proportional maximin share) allocations for indivisible goods. Focusing on three structured valuation classes—personalized binary valuations, binary MMS-feasible valuations, and pairwise-demand valuations—the authors employ constructive proofs and polynomial-time algorithm design. Their contributions are threefold: (1) They construct the first explicit counterexample showing that a PMMS allocation may not exist for three agents, thereby strictly separating PMMS from EFX; (2) They prove that an EFX allocation always exists under personalized binary valuations and provide an efficient constructive algorithm; (3) They extend the existence guarantees of PMMS to chores and mixed goods-chore settings, establishing tractability under specific conditions. Collectively, these results precisely delineate the theoretical boundaries of EFX and PMMS, advancing the feasibility theory of discrete fair division.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous 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
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We study the fair division of indivisible items and provide new insights into the EFX problem, which is widely regarded as the central open question in fair division, and the PMMS problem, a strictly stronger variant of EFX. Our first result constructs a three-agent instance with two monotone valuations and one additive valuation in which no PMMS allocation exists. Since EFX allocations are known to exist under these assumptions, this establishes a formal separation between EFX and PMMS. We prove existence of fair allocations for three important special cases. We show that EFX allocations exist for personalized bivalued valuations, where for each agent $i$ there exist values $a_i > b_i$ such that agent $i$ assigns value $v_i({g}) in {a_i, b_i}$ to each good $g$. We establish an analogous existence result for PMMS allocations when $a_i$ is divisible by $b_i$. We also prove that PMMS allocations exist for binary-valued MMS-feasible valuations, where each bundle $S$ has value $v_i(S) in {0, 1}$. Notably, this result holds even without assuming monotonicity of valuations and thus applies to the fair division of chores and mixed manna. Finally, we study a class of valuations called pair-demand valuations, which extend the well-studied unit-demand valuations to the case where each agent derives value from at most two items, and we show that PMMS allocations exist in this setting. Our proofs are constructive, and we provide polynomial-time algorithms for all three existence results.

Problem

• Study EFX and PMMS in indivisible item fair division
• Prove existence of EFX for personalized bivalued valuations
• Establish PMMS existence for binary-valued MMS-feasible valuations

Innovation

• Constructs three-agent instance for PMMS separation
• Proves EFX existence for personalized bivalued valuations
• Establishes PMMS existence for binary-valued MMS-feasible valuations