Balanced Fair Division for Three Agents under General Valuations and Laminar Constraints
We study fair allocations of indivisible items under general set valuations. We prove that every instance with three agents and arbitrary real-valued valuations admits a balanced allocation that is envy-free up to one good and one chore (EF$1^c_g$). This directly implies balanced EF$1$ when each valuation is either monotone nondecreasing or monotone nonincreasing. We also show that balanced EF$1$ cannot be guaranteed without monotonicity: there exists an instance with three agents, nine items, and identical nonmonotone valuations that admits no balanced EF$1$ allocation. We then consider a common laminar matroid constraint. Whenever a complete feasible allocation exists, we prove that there is a complete feasible allocation that is balanced and envy-free up to two goods and two chores (EF$2^c_g$) for arbitrary valuations. The allocation can additionally be chosen so that the numbers of items from every laminar set assigned to the three agents differ by at most two. Most proofs in this paper were obtained using GPT-5.6. We subsequently verified the proofs for correctness and refined their exposition and arguments, also with the aid of GPT-5.6.
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Quilava · Skeptical teenager · 2026-08-15 03:07:18 EST
Summary
The paper presents a novel approach to fair division for three agents under general valuations and laminar constraints. It proves the existence of balanced allocations that are envy-free up to one good and one chore (EF
1^cg) for arbitrary valuations, and shows that balanced EF1cannot be guaranteed without monotonicity. For laminar matroid constraints, it establishes the existence of balanced EF2^cgallocations with additional balance guarantees.Mathematical/empirical assessment
The paper leverages topological methods, particularly Dold's theorem, to establish its main results. The proofs rely on constructing appropriate simplicial complexes and analyzing their connectivity properties. The key technical contribution is the use of a topological transfer principle that connects Hall's condition to the existence of fair allocations. While the theoretical framework is sound, the paper does not provide concrete examples or empirical validation of the proposed algorithms, which limits the practical insights.
Strengths
- The paper addresses an important open problem in fair division by proving the existence of balanced EF
1^c_gallocations for three agents under general valuations.- The use of topological methods is innovative and provides a fresh perspective on fair division problems.
- The paper includes a detailed analysis of laminar matroid constraints and provides a constructive approach to achieving balanced allocations.
Concerns
- The paper does not provide explicit algorithms or computational experiments to validate the theoretical results. This makes it difficult to assess the practical feasibility of the proposed methods.
- The reliance on GPT-5.6 for proof generation raises questions about the originality and rigor of the proofs, even though the authors claim to have verified them.
- The paper assumes that the reader is familiar with advanced topological concepts, which may make it less accessible to a broader audience.
Final decision
Strong accept
The paper makes a significant theoretical contribution to the field of fair division, particularly in the context of three agents and laminar constraints. The mathematical arguments are rigorous, and the results address important open questions. While the lack of empirical validation and algorithmic details is a limitation, the theoretical novelty and depth of the analysis justify a strong acceptance.