Learning Fair Allocation of Indivisible Items from Limited Feedback

2026-06-30

AI Summary

This work addresses the challenge of online learning fair allocations of indivisible goods under limited and potentially adversarial fairness feedback. Focusing on achieving EF1 or PROP1 fairness criteria, the paper proposes a general framework that integrates the geometry of valuation polytopes with the ellipsoid method, leveraging separation hyperplanes to handle unknown agent preferences. The authors establish that an EF1 allocation necessarily exists under the “interval plus one” preference structure. Their approach combines polytope maintenance, robust handling of adversarial feedback, and structured search to compute EF1 or PROP1 allocations in polynomial time for additive valuations. Furthermore, for general monotone valuations, the algorithm converges to an EF1 allocation within a polynomial number of rounds.

Abstract

We study a setting in which an algorithm must output a fair allocation of indivisible items while "learning on the job". More specifically, the algorithm is to output an allocation satisfying EF1, PROP1, or similar fairness notions; however, the algorithm initially has no information about the agents' valuations, and can only learn about them by (repeatedly) proposing an allocation, and obtaining feedback about a fairness violation in the allocation. Importantly, the observed fairness violation may be adversarially chosen. The algorithm's goal is to converge to a fair allocation in rounds polynomial in the number of agents and items, ideally with only polynomial computation. We prove two main results: first, when the valuations are additive, then even for mixed items (goods and chores), an allocation satisfying EF1 or PROP1 can be found in polynomial time using the corresponding feedback. These results are instantiations of a more general framework which maintains a polytope of candidate valuations consistent with all past feedback. The algorithm repeatedly constructs putative valuations and uses them to propose allocations; the observed violations then define separating hyperplanes, allowing the algorithm to emulate the ellipsoid method. When the valuations are monotone, we present an algorithm which is guaranteed to find an EF1 allocation in polynomially many iterations; however, its internal calculations are not guaranteed to be polynomial. The algorithm again maintains putative valuations, and only considers allocations in which each agent obtains an interval plus one additional item with respect to an arbitrary ordering of the items. We (non-constructively) prove that there always exist EF1 allocations of this form, allowing us to use a further generalization of the preceding ellipsoid-based ideas.

Problem

• fair allocation
• indivisible items
• limited feedback
• EF1
• PROP1

Innovation

• fair allocation
• limited feedback
• ellipsoid method
• EF1
• indivisible items