Best-of-Both-Worlds Fairness for Mixed Goods and Chores

2026-07-11

AI Summary

This study addresses the problem of fair random allocation of indivisible mixed goods—items that are desirable to some agents and undesirable to others—with the dual objective of achieving ex-ante envy-freeness (EF) and ex-post envy-freeness up to one item (EF1). To this end, the authors introduce a novel probabilistic Hall-type matrix decomposition technique, which integrates Sion’s minimax theorem with biased flow network methods. This approach enables the first known mechanism that simultaneously guarantees ex-ante EF and ex-post EF1 in settings involving mixed items, thereby resolving a central open challenge in the fair division literature concerning the compatibility of these two fairness criteria under randomness.

Abstract

We study the fundamental problem of fairly dividing indivisible items among agents with additive utilities. In our model, an item can be a good yielding non-negative utilities to some agents and simultaneously a chore yielding negative utilities to others. We take the best-of-both-worlds perspective and our goal is to construct a randomized allocation that is exactly fair ex ante while also being supported on ex post approximately fair allocations. The fairness notions examined in this paper are envy-freeness (EF) and its well-known relaxation envy-freeness up to one item (EF1). Our main result is that ex-ante EF and ex-post EF1 can be achieved simultaneously. To achieve this, we introduce a novel probabilistic Hall-type matrix decomposition that intricately correlates the fractional assignments of goods and chores. We resolve this decomposition problem by combining continuous minimax duality -- via Sion's minimax theorem -- with carefully designed biased flow networks.

Problem

• fair division
• mixed goods and chores
• envy-freeness
• EF1
• indivisible items

Innovation

• best-of-both-worlds fairness
• mixed goods and chores
• ex-ante EF
• ex-post EF1
• probabilistic Hall-type decomposition