Fair allocation of a multiset of indivisible items

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 $t\le2$, 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.