Existence of MMS Allocations with Mixed Manna

Abstract: Maximin share (MMS) allocations are a popular relaxation of envy-free allocations that have received wide attention in the fair division of indivisible items. Although MMS allocations of goods can fail to exist, previous work has found conditions under which they exist. Specifically, MMS allocations of goods exist whenever $m \leq n+5$, and this bound is tight in the sense that they can fail to exist when $m = n+6$. The techniques used to obtain these results do not apply to the mixed manna setting, leaving the question of whether similar results hold for the general setting. This paper addresses this by introducing new techniques to handle these settings. In particular, we are able to answer this question completely for the chores setting, and partially for the mixed manna setting. An agent $i$ is a {\em chores agent} if it considers every item to be a chore and a {\em non-negative agent} if its MMS guarantee is non-negative. In this paper, we prove that an MMS allocation exists as long as $m \leq n+5$ and either (i) every agent is a chores agent, or (ii) there exists a non-negative agent. In addition, for $n \leq 3$, we also prove that an MMS allocation exists as long as $m \leq n+5$, regardless of the types of agents. To the best of our knowledge, these are the first non-trivial results pertaining to the existence of exact MMS allocations in the mixed manna setting.