Existence of EFX for Two Additive Valuations
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Fair division of indivisible items is a well-studied topic in Economics and Computer Science. The objective is to allocate items to agents in a fair manner, where each agent has a valuation for each subset of items. Several concepts of fairness have been established, and envy-freeness is one of the most widely studied notions of fairness. Since envy-free allocations do not always exist when items are indivisible, several relaxations have been considered. Among them, possibly the most compelling one is envy-freeness up to any item (EFX), where no agent envies another agent after the removal of any single item from the other agent's bundle. However, despite significant efforts by many researchers for several years, it is only known that an EFX allocation always exists when there are at most three agents and their valuations are additive or when all agents have identical valuations. In this paper, we show that an EFX allocation always exists when every agent has one of the two additive valuations. We give a constructive proof, in which we iteratively obtain a Pareto dominating EFX allocation from an existing EFX allocation.
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