Best of Both Worlds Guarantees for Equitable Allocations

2025-05-09

AI Summary

This paper studies “best-of-both-worlds” fairness in randomized allocation of indivisible goods: requiring strict ex ante envy-freeness (EQ) and ex post envy-freeness up to one good (EQ1) for all deterministic outcomes in the support. First, it provides a systematic geometric characterization—necessary and sufficient conditions—for the existence of such allocations. It proves that for two agents, such allocations always exist and are constructible in polynomial time; for three or more agents, existence testing is NP-complete, yet a pseudopolynomial-time algorithm is provided. Under binary valuations, it further achieves simultaneous EQ1 and social welfare maximization. The core contribution lies in unifying the feasibility boundaries of ex ante and ex post fairness, and establishing tight complexity classifications alongside efficient algorithms.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Equitability is a well-studied fairness notion in fair division, where an allocation is equitable if all agents receive equal utility from their allocation. For indivisible items, an exactly equitable allocation may not exist, and a natural relaxation is EQ1, which stipulates that any inequitability should be resolved by the removal of a single item. In this paper, we study equitability in the context of randomized allocations. Specifically, we aim to achieve equitability in expectation (ex ante EQ) and require that each deterministic outcome in the support satisfies ex post EQ1. Such an allocation is commonly known as a `Best of Both Worlds' allocation, and has been studied, e.g., for envy-freeness and MMS. We characterize the existence of such allocations using a geometric condition on linear combinations of EQ1 allocations, and use this to give comprehensive results on both existence and computation. For two agents, we show that ex ante EQ and ex post EQ1 allocations always exist and can be computed in polynomial time. For three or more agents, however, such allocations may not exist. We prove that deciding existence of such allocations is strongly NP-complete in general, and weakly NP-complete even for three agents. We also present a pseudo-polynomial time algorithm for a constant number of agents. We show that when agents have binary valuations, best of both worlds allocations that additionally satisfy welfare guarantees exist and are efficiently computable.

Problem

• Achieving equitable allocations in randomized fair division
• Characterizing existence of ex ante EQ and ex post EQ1 allocations
• Computational complexity of equitable allocations for multiple agents

Innovation

• Ex ante EQ and ex post EQ1 allocations
• Geometric condition on EQ1 allocations
• Pseudo-polynomial time algorithm