We study the problem of fairly allocating indivisible goods when limited sharing is allowed, that is, each good may be allocated to up to $k$ agents, while incurring a cost for sharing. While classic maximin share (MMS) allocations may not exist in many instances, we demonstrate that allowing controlled sharing can restore fairness guarantees that are otherwise unattainable in certain scenarios. (1) Our first contribution shows that exact maximin share (MMS) allocations are guaranteed to exist whenever goods are allowed to be cost-sensitively shared among at least half of the agents and the number of agents is even; for odd numbers of agents, we obtain a slightly weaker MMS guarantee. (2) We further design a Shared Bag-Filling Algorithm that guarantees a $(1 - C)(k - 1)$-approximate MMS allocation, where $C$ is the maximum cost of sharing a good. Notably, when $(1 - C)(k - 1) \geq 1$, our algorithm recovers an exact MMS allocation. (3) We additionally introduce the Sharing Maximin Share (SMMS) fairness notion, a natural extension of MMS to the $k$-sharing setting. (4) We show that SMMS allocations always exist under identical utilities and for instances with two agents. (5) We construct a counterexample to show the impossibility of the universal existence of an SMMS allocation. (6) Finally, we establish a connection between SMMS and constrained MMS (CMMS), yielding approximation guarantees for SMMS via existing CMMS results. These contributions provide deep theoretical insights for the problem of fair resource allocation when a limited sharing of resources are allowed in multi-agent environments.

翻译：本文研究了在允许有限共享条件下不可分割物品的公平分配问题，即每个物品最多可分配给$k$个智能体，但共享会产生成本。尽管经典的最大最小份额（MMS）分配在许多情况下可能不存在，我们证明允许受控共享能够恢复某些场景下原本无法实现的公平性保证。（1）我们的第一个贡献表明：当允许物品在至少半数智能体之间进行成本敏感共享且智能体数量为偶数时，精确的最大最小份额（MMS）分配必然存在；对于奇数个智能体，我们获得了稍弱的MMS保证。（2）我们进一步设计了共享装袋算法，该算法可保证实现$(1 - C)(k - 1)$-近似MMS分配，其中$C$为物品共享的最大成本。值得注意的是，当$(1 - C)(k - 1) \geq 1$时，我们的算法可恢复精确的MMS分配。（3）我们额外提出了共享最大最小份额（SMMS）公平性概念，这是MMS在$k$共享设置下的自然扩展。（4）我们证明在效用函数相同的情况下以及双智能体场景中，SMMS分配始终存在。（5）我们构造了反例以证明SMMS分配普遍存在的不可能性。（6）最后，我们建立了SMMS与约束最大最小份额（CMMS）之间的联系，通过现有CMMS结果获得了SMMS的近似保证。这些贡献为多智能体环境中允许资源有限共享的公平资源分配问题提供了深刻的理论见解。