GPT-5.6-Sol helps prove fair allocation theorem for indivisible goods

In a new arXiv paper, computer scientists Benjamin Cookson, Nisarg Shah, and Paritosh Verma tackle a fundamental question in fair division: can indivisible goods be allocated fairly and efficiently when bundles must be balanced in size? They prove that for any additive valuations—where each good's value is simply summed—there always exists a balanced allocation that is both envy-free up to one good (EF1) and fractionally Pareto optimal (fPO). This generalizes a 2026 result by Kawase et al., which only covered personalized bivalued valuations or at most two distinct valuation types.

The proof leverages the Knaster-Kuratowski-Mazurkiewicz (KKM) lemma within a weighted-welfare duality framework, overcoming barriers with a novel price-interlacing lemma. The team also extends the technique to category constraints (partition-matroid constraints), establishing existence of fPO allocations under a weaker category-sensitive EF1 relaxation. Notably, all proofs in the paper were generated using GPT-5.6-Sol, with guidance from the authors, who verified and simplified the arguments with assistance from GPT-5.6-Sol and Claude Fable 5—a striking example of AI-accelerated theoretical research.