We study randomized allocations (lotteries) of indivisible chores under additive costs and introduce the notion of MMS1, which requires each agent's bundle to satisfy the maximin share (MMS) bound after the removal of at most one chore. Our main result establishes that every instance admits an ex-ante proportional lottery in whose support every allocation is MMS1 and all but $o(n)$ agents satisfy MMS. Consequently, every realized allocation can be made MMS by disposing of at most $o(n)$ chores. The proposed framework further achieves ex-ante proportionality alongside ex-post $4/3$-MMS, $o(n)$-AMMS, or $1$-out-of-$d$ MMS for $d=(1-o(1))n$. These guarantees substantially improve upon state-of-the-art bounds for MMS allocation for chores with disposal or subsidy, best-of-both-worlds fairness, approximate AMMS, and ordinal MMS approximations.
Web platforms organize collective work, including content review and community maintenance, that imposes heterogeneous effort costs on participants. Fairly distributing these burdens while leaving as few tasks unassigned as possible is a natural objective for the design of such platforms. We study this through the maximin share (MMS) allocation problem with chore disposals under nonnegative additive costs. Our goal is to minimize the number of unassigned chores while guaranteeing every agent her MMS benchmark from the original instance. We prove that, for every instance with $n \ge 4$ agents, at most $n-3$ chore disposals suffice. We improve the bound of $2m/11+n$ disposals established by Barman et al. (2025), where $m$ is the number of chores, and eliminate the dependence on $m$. For four agents, we obtain a tight bound: one disposal always suffices, whereas known instances admit no complete MMS allocation (Ezra and Garbuz, 2026). We also prove the same result for goods: $n-3$ duplications suffice to ensure MMS.