Maximin share (MMS) stands out as a central notion in fair resource allocation. It is known that exact MMS fairness is not always attainable, especially when agents differ along two dimensions: their valuations and their perceptions of the divisibility of resources. The former case with heterogeneous valuations has been widely studied in the literature. The latter, referred to as subjective divisibility by Bei et al., [Games Econ. Behav. 2025], remains much less explored. We study MMS approximation under subjective divisibility. First, we prove that even in the unary valuation setting, where all items have equal value, the optimal approximation ratio is 2/3. This result is somewhat surprising since in the objective setting, even when agents have heterogeneous valuations, the best possible approximation ratio is at least 7/9 [Huang and Zhou, 2025]. We then address the general case with both valuation heterogeneity and subjective divisibility. Previous work shows the existence of a 1/2-approximate MMS allocation. In this paper, we develop new algorithmic techniques that overcome the difficulties posed by subjective divisibility, and improve the approximation guarantee to 5/9. Finally, we complement this result with small-agent cases. For up to four agents, we give polynomial-time algorithms that compute 2/3-approximate MMS fair allocations. These bounds are tight. Our results deepen the understanding of MMS fairness under heterogeneous valuations and subjective divisibility, and provide a new perspective for this emerging model.

翻译：最大最小份额是公平资源分配中的核心概念。已知精确的MMS公平性并非总能实现，尤其是当主体在估值和资源可分性感知两个维度存在差异时。前者（异质估值）在文献中已有广泛研究，而后者（即Bei等人提出的"主观可分性"，[Games Econ. Behav. 2025]）的探索仍相对有限。我们研究主观可分性下的MMS近似问题。首先证明：即便在单项估值相同的一元估值设置中，最优近似比也为2/3。此结果出人意料——因为在客观设置中，即便主体具有异质估值，最优近似比仍可达至少7/9 [Huang and Zhou, 2025]。我们进一步研究估值异质性与主观可分性并存的通用情形。既有工作已证明1/2近似MMS分配的存在性，本文开发了能克服主观可分性难题的新算法技术，将近似保证提升至5/9。最后，我们针对小型主体情形完善该结果：针对最多四个主体的情况，给出了多项式时间算法计算2/3近似MMS公平分配，且该界是紧的。这些研究深化了对异质估值与主观可分性下MMS公平性的理解，并为这一新兴模型提供了新视角。