EFX Exists for Three Agents
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We study the problem of allocating a set of indivisible items among agents with additive valuations in a fair manner. Envy-freeness up to any item (EFX) is arguably the most compelling fairness concept for this problem. However, despite significant efforts by many researchers for several years, its existence has not been settled beyond the simple case of two agents. In this paper, we break this barrier by showing that an EFX allocation always exists for three agents! Our proof is algorithmic and quite involved. Furthermore, we also falsify a conjecture of Caragiannis et al. by showing an instance with three agents for which there is a partial EFX allocation (some items are not allocated) with higher Nash welfare than that of any complete EFX allocation.
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