Fair allocation of a multiset of indivisible items
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We study the problem of allocating a set M of m indivisible items among n agents in a fair manner. We consider two well-studied notions of fairness: envy-freeness (EF), and envy-freeness up to any good (EFX). While it is known that complete EF allocations do not always exist, it is not known if complete EFX allocations exist besides a few cases. In this work, we reformulate the problem to allow M to be a multiset. Specifically, we introduce a parameter t for the number of distinct types of items, and study allocations of multisets that contain items of these t types. We show the following: 1. For arbitrary n, t, a complete EF allocation exists when agents have distinct additive valuations, and there are enough items of each type. 2. For arbitrary n, m, t, a complete EFX allocation exists when agents have additive valuations with identical preferences. 3. For arbitrary n, m, and t≤2, a complete EFX allocation exists when agents have additive valuations. For 2 and 3, our approach is constructive; we give a polynomial-time algorithm to find a complete EFX allocation.
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