On Fair Allocation of Indivisible Goods to Submodular Agents

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 13-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 13 of their APS. For the equal entitlement case, we improve the approximation ratio and obtain 1027-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.