We consider the problem of fair allocation of indivisible goods to agents with submodular valuation functions, where agents may have either equal entitlements or arbitrary (possibly unequal) entitlements. We focus on share-based fairness notions, specifically, the maximin share (MMS) for equal entitlements and the anyprice share (APS) for arbitrary entitlements, and design allocation algorithms that give each agent a bundle of value at least some constant fraction of her share value. For the equal entitlement case (and submodular valuations), Ghodsi, Hajiaghayi, Seddighin, Seddighin, and Yami [EC 2018] designed a polynomial-time algorithm for $\frac{1}{3}$-maximin-fair allocation. We improve this result in two different ways. We consider the general case of arbitrary entitlements, and present a polynomial time algorithm that guarantees submodular agents $\frac{1}{3}$ of their APS. For the equal entitlement case, we improve the approximation ratio and obtain $\frac{10}{27}$-maximin-fair allocations. Our algorithms are based on designing strategies for a certain bidding game that was previously introduced by Babaioff, Ezra and Feige [EC 2021].

翻译：我们考虑将不可分物品公平分配给具有次模估值函数的智能体的问题，其中智能体可能拥有相等的权利或任意（可能不相等的）权利。我们重点关注基于份额的公平概念，具体而言，针对相等权利的最大最小份额（MMS），以及针对任意权利的任意价格份额（APS），并设计分配算法，为每个智能体提供价值至少为其份额值某个常数倍的一束物品。针对相等权利情况（及次模估值），Ghodsi、Hajiaghayi、Seddighin、Seddighin 和 Yami [EC 2018] 设计了一个多项式时间算法，用于实现$\frac{1}{3}$-最大最小公平分配。我们从两个方面改进了这一结果。我们考虑了任意权利的一般情况，并提出了一个多项式时间算法，保证次模智能体获得其APS的$\frac{1}{3}$。对于相等权利情况，我们改进了近似比率，获得了$\frac{10}{27}$-最大最小公平分配。我们的算法基于为先前由Babaioff、Ezra和Feige [EC 2021] 提出的某种竞价博弈设计策略。