Maximin share (MMS) allocations are a popular relaxation of envy-free allocations that have received wide attention in the context of the fair division of indivisible items. Although MMS allocations can fail to exist [1], previous work has found conditions under which they exist. Specifically, MMS allocations exist whenever $m \leq n+5$ in the context of goods allocation, and this bound is tight in the sense that MMS allocations can fail to exist when $m = n+6$ [2]. Unfortunately, the technique used to establish this result does not generalize readily to the chores and mixed manna settings. This paper generalizes this result to the chores setting and provides a partial solution for the mixed manna setting. Our results depend on the presence of certain types of agents. Specifically, an agent $i$ is a goods agent (resp. chores agent) if every item is a good (resp. chore) to $i$, and a non-negative mixed agent if $i$ is neither a goods nor a chores agent and the MMS guarantee of $i$ is non-negative. In this paper, we prove that an MMS allocation exists if $m \leq n+5$ and there exists a goods agent, a non-negative mixed agent, or only chores agents. [1] David Kurokawa, Ariel D Procaccia, and Junxing Wang. When can the maximin share guarantee be guaranteed? In Thirtieth AAAI Conference on Artificial Intelligence, 2016. [2] Uriel Feige, Ariel Sapir, and Laliv Tauber. A tight negative example for mms fair allocations. In International Conference on Web and Internet Economics, pages 355-372. Springer, 2021.

翻译：最大最小份额（MMS）分配是嫉妒无分配的一种流行松弛，在不可分物品公平分配领域受到广泛关注。尽管MMS分配可能不存在[1]，但已有研究发现了其存在的条件。具体而言，在物品分配场景中，当$m \leq n+5$时，MMS分配必然存在，且该界限是紧的——当$m = n+6$时，MMS分配可能不存在[2]。遗憾的是，用于建立该结果的方法无法直接推广到杂务分配和混合物品场景中。本文将结果推广至杂务分配场景，并给出混合物品场景的部分解。我们的结论依赖于特定类型智能体的存在。具体而言，若所有物品对智能体$i$均为“好”（good），则称其为“物品型智能体”（goods agent）；若所有物品对智能体$i$均为“坏”（chore），则称其为“杂务型智能体”（chores agent）；若智能体$i$既非物品型也非杂务型，且其MMS保障值为非负，则称其为“非负混合型智能体”（non-negative mixed agent）。本文证明：当$m \leq n+5$且存在至少一个物品型智能体、非负混合型智能体，或仅包含杂务型智能体时，MMS分配存在。[1] David Kurokawa, Ariel D Procaccia, and Junxing Wang. When can the maximin share guarantee be guaranteed? In Thirtieth AAAI Conference on Artificial Intelligence, 2016. [2] Uriel Feige, Ariel Sapir, and Laliv Tauber. A tight negative example for mms fair allocations. In International Conference on Web and Internet Economics, pages 355-372. Springer, 2021.