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arXiv · 2202.05186

Fair allocation of a multiset of indivisible items

Pranay Gorantla·Kunal Marwaha·Santhoshini Velusamy
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Abstract

We study the problem of fairly allocating a multiset $M$ of $m$ indivisible items among $n$ agents with additive valuations. Specifically, we introduce a parameter $t$ for the number of distinct types of items and study fair allocations of multisets that contain only items of these $t$ types, under two standard notions of fairness: 1. Envy-freeness (EF): For arbitrary $n$, $t$, we show that a complete EF allocation exists when at least one agent has a unique valuation and the number of items of each type exceeds a particular finite threshold. We give explicit upper and lower bounds on this threshold in some special cases. 2. Envy-freeness up to any good (EFX): For arbitrary $n$, $m$, and for $t\le 2$, we show that a complete EFX allocation always exists. We give two different proofs of this result. One proof is constructive and runs in polynomial time; the other is geometrically inspired.

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Pranay Gorantla, Kunal Marwaha, Santhoshini Velusamy. 2022-02-10. Fair allocation of a multiset of indivisible items. https://doi.org/10.1137/1.9781611977554.ch13

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