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.
We study the comparison-query complexity of computing allocations that are envy-free up to one item (EF1) for $n$ agents with heterogeneous additive preferences over $m$ indivisible items. In this model, the algorithm may only query which of two bundles an agent prefers, without ever observing numerical valuations or costs. Prior work established $O(\log m)$ query bounds for at most three agents, leaving open whether query complexity logarithmic in $m$ is achievable for every fixed $n \ge 4$. We resolve this open question in the affirmative: for every fixed $n\ge2$, we design algorithms that compute EF1 allocations using $O(\log m)$ comparison queries, matching the known $Ω(\log m)$ lower bound. Our results hold for both chores with nonnegative costs and goods with nonnegative valuations.
We study gains from trade (GFT) in two-sided matching markets with independent private types and arbitrary downward-closed feasibility constraints. The second-best benchmark is the maximum expected GFT achievable by a Bayesian incentive-compatible, interim individually rational mechanism that is strongly budget balanced at every report profile. These constraints generally preclude attaining the first-best GFT and raise the question of how much efficiency must be lost. We prove that the second-best GFT is at least one half of the first-best GFT in every such matching market. This recovers and generalizes the recent $1/2$ guarantee for bilateral trade by Liu, Qin, Ren, and Wang to markets with multiple buyers and sellers and arbitrary downward-closed feasibility constraints. Together with their matching upper bound for bilateral trade, our result establishes a tight worst-case ratio of $1/2$ for this general class of matching markets. Our proof builds on the virtual-GFT framework of Brüstle, Cai, Wu, and Zhao to reduce the problem to a one-parameter Lagrangian. The main step is a geometric, edge-by-edge analysis based on first-best edge-selection regions, combined with a randomized contraction in rank space. The same analysis also determines the exact worst-case efficiency for every level of outside subsidy. In particular, even a small outside subsidy can significantly improve the worst-case guarantee.