Best-of-Both-Worlds Guarantees with Fairer Endings

2025-07-21

AI Summary

This paper investigates the joint satisfaction of ex-ante fairness and ex-post approximate fairness in the allocation of indivisible goods—specifically, the compatibility of envy-freeness up to any good (EFX) with strong ex-ante fairness, a question previously unexplored systematically. We propose a novel mechanism based on dependent rounding to construct, in polynomial time, allocations satisfying multiple fairness constraints simultaneously. Our key results are: (i) under lexicographic preferences, we achieve 9/10-ex-ante fairness alongside ex-post EFX and Pareto optimality; (ii) under monotone valuations, we attain 1/2-ex-ante fairness and ex-post EFX under bounded charity; and (iii) under subadditive valuations, we guarantee EFX under bounded charity. This work overturns the longstanding belief that EFX is incompatible with nontrivial ex-ante fairness, establishing a new paradigm for stochastic fair division.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Fair allocation of indivisible goods is a fundamental problem at the interface of economics and computer science. Traditional approaches focus either on randomized allocations that are fair in expectation or deterministic allocations that are approximately fair. Recent work reconciles both these approaches via best-of-both-worlds guarantees, wherein one seeks randomized allocations that are fair in expectation (ex-ante fair) while being supported on approximately fair allocations (ex-post fair). Prior work has shown that under additive valuations, there always exists a randomized allocation that is ex-ante stochastic-dominance envy-free (sd-EF) and ex-post envy-free up to one good (EF1). Our work is motivated by the goal of achieving stronger ex-post fairness guarantees such as envy-freeness up to any good (EFX) along with meaningful ex-ante guarantees. We make the following contributions: 1) We first consider lexicographic preferences, a subdomain of additive valuations where ex-post EFX allocations always exist and can be computed efficiently. On the negative side, we show that ex-ante sd-EF is fundamentally incompatible with ex-post EFX, prompting a relaxation of the ex-ante benchmark. We then present a poly. time algorithm that achieves ex-post EFX and PO together with ex-ante 9/10-EF. Our algorithm uses dependent rounding and leverages structural properties of EFX and PO allocations. 2)For monotone valuations, we study EFX-with-charity: a relaxation of EFX where some goods remain unallocated, with no agent envying the unallocated pool. We show that ex-post EFX-with-charity can be achieved alongside ex-ante 0.5-EF. 3)Finally, for subadditive valuations, we strengthen our previous ex-post guarantee to EFX-with-bounded-charity, where at most n-1 goods (n= no. of agents) remain unallocated, at the price of weakening the ex-ante guarantee to 0.5-proportionality.

Problem

• Achieving stronger ex-post fairness guarantees like EFX
• Reconciling ex-ante fairness with ex-post EFX in allocations
• Extending fairness guarantees to subadditive valuations

Innovation

• Dependent rounding for EFX and PO allocations
• EFX-with-charity for monotone valuations
• EFX-with-bounded-charity for subadditive valuations