Epistemic fair division of independence structures

2026-06-09

AI Summary

This study addresses the fair allocation of indivisible goods under independence constraints—such as those defined by matroids or acyclic graphs—when agents possess arbitrary additive valuations. It introduces a novel fairness notion, “cognitive EF1,” requiring that each agent receives a feasible bundle and that there exists a feasible partition of the remaining items such that the agent envies no other agent by more than one item. This work pioneers the integration of fair division with abstract independence systems, establishing the universal existence of cognitive EF1 allocations under general independence structures. It further connects this guarantee to deep conjectures in matroid theory—for instance, yielding an EF1 partitioning algorithm for bipartite Hamiltonian matroids. Leveraging tools from combinatorial optimization, matroid theory, and graph theory (including treewidth and forest decompositions), the authors devise efficient constructive algorithms and validate their approach through concrete examples.

Abstract

We study the problem of fair division of indivisible goods with constraints imposed by a prescribed independence structure, that is, a family of subsets of goods closed under taking subsets. As a motivating example, imagine that the goods to be divided are the available connections in a logistic, financial, or social network. The admissible bundle of goods for each agent must correspond to an acyclic set of edges, corresponding to a basic feasible solution to a linear network problem to be solved. Suppose that all agents assign the same value to each good (in the example, the network connections are equally important for every agent) and evaluate each bundle by summing the values of its goods. Is there a fair partition of the goods into such acyclic bundles? Surprisingly, the answer is yes, provided that the number of agents is at least the arboricity of $G$, and the fairness requirement is envy-freeness up to one good (EF1). The situation becomes more mysterious when agents have arbitrary additive valuations. Our main result guarantees that, in this case, epistemic EF1 partitions always exist, which means that each agent receives an acyclic bundle for which there exists a feasible partition of the remaining goods into acyclic bundles that they do not envy up to one good. We derive this conclusion from a general result for abstract independence structures defined on the sets of goods. We also discuss connections with several conjectures concerning matroids. In particular, we prove that any Hamiltonian matroid partitionable into two independent sets admits an EF1 bipartition with respect to a common monotone valuation. We complement our results with a constructive perspective: we present explicitly two algorithms for computing the fair allocations described above. Finally, we provide illustrative examples to demonstrate these algorithms on specific instances.

Problem

• fair division
• independence structures
• indivisible goods
• envy-freeness up to one good
• matroids

Innovation

• epistemic EF1
• independence structures
• fair division
• matroid partition
• constructive algorithms