We consider the problem of allocating $m$ indivisible chores to $n$ agents with additive disvaluation (cost) functions. It is easy to show that there are picking sequences that give every agent (that uses the greedy picking strategy) a bundle of chores of disvalue at most twice her share value (maximin share, MMS, for agents of equal entitlement, and anyprice share, APS, for agents of arbitrary entitlement). Aziz, Li and Wu (2022) designed picking sequences that improve this ratio to $\frac{5}{3}$ for the case of equal entitlement. We design picking sequences that improve the ratio to~1.733 for the case of arbitrary entitlement, and to $\frac{8}{5}$ for the case of equal entitlement. (In fact, computer assisted analysis suggests that the ratio is smaller than $1.543$ in the equal entitlement case.) We also prove a lower bound of $\frac{3}{2}$ on the obtainable ratio when $n$ is sufficiently large. Additional contributions of our work include improved guarantees in the equal entitlement case when $n$ is small; introduction of the chore share as a convenient proxy to other share notions for chores; introduction of ex-ante notions of envy for risk averse agents; enhancements to our picking sequences that eliminate such envy; showing that a known allocation algorithm (not based on picking sequences) for the equal entitlement case gives each agent a bundle of disvalue at most $\frac{4n-1}{3n}$ times her APS (previously, this ratio was shown for this algorithm with respect to the easier benchmark of the MMS).
翻译:我们考虑的是将美元不可分割的杂务分配给具有累加性贬值(成本)功能的代理商的问题。我们很容易地看到,有选择顺序,让每个代理商(使用贪婪的挑剔策略)的杂务价值最多为其份额价值的两倍(最大份额,MMS,用于同等应享权利代理商,和任何价格份额,APS,APS)。Aziz、Li和Wu(2022年)设计了将这一比率提高到美元比额的顺序,用于同等待遇。我们设计了将这一比率提高到美元比额的顺序,用于提高美元比额的比额。我们设计了将任意应享待遇比额比重提高到~1.733的顺序,使每个代理商的比重提高到1美元;在每份平分法中引入一个比值比例,用于平分比值比值的顺序。