Balanced Fair Division for Three Agents under General Valuations and Laminar Constraints

Abstract

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.