Year: 2026
Type: preprint
Abstract: We show that every additive chore instance with three agents and seven or eight indivisible chores has an EFX allocation. Costs may be arbitrary nonnegative reals. Together with the theorem of Kobayashi, Mahara, and Sakamoto for at most $2n$ chores, this settles the existence of EFX allocations for three agents and at most eight chores; for four or more agents, He and Tao have shown that EFX allocations of additive chores need not exist. Both proofs are computer-assisted. For seven chores, we normalize each agent's costs and sort the chores, and then express the nonexistence of an EFX allocation as a formula of quantifier-free linear real arithmetic with one clause per allocation; the SMT solvers Z3 and cvc5 show that this formula is unsatisfiable. For eight chores, if two agents share a cheapest chore, we remove that chore, apply the seven-chore result, and put the chore back using an insertion lemma of Kobayashi, Mahara, and Sakamoto. The remaining instances, in which the agents' sets of cheapest chores are pairwise disjoint, are handled by a second formula of the same kind.
Source: arXiv (Cornell University)
Author: Zhang, Xinkai
Institution: Renmin University of China
Language: English
Subfield: Economics and Econometrics
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W7212302698