Fair allocation of a multiset of indivisible items

TL;DR

This paper investigates fair division of multisets of indivisible items among agents, establishing existence results for envy-freeness and EFX under various conditions, with some algorithms provided.

Contribution

It introduces new existence results for envy-freeness and EFX in multiset allocations, including polynomial-time algorithms for specific cases.

Findings

EF allocations exist when at least one agent has a unique valuation and item counts are sufficiently large.

EFX allocations always exist for up to two item types, with constructive and geometric proofs.

Explicit bounds and thresholds for fair allocations are provided in special cases.

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.