与室友共同生活时，清洁问题常引发矛盾。无论是室友邋遢自己需多做清洁，还是室友对清洁标准要求过高，都会导致一方承担过多清洁任务。文章指出，简单的平均分担或一方妥协都难以实现公平。作者引入了Shapley值概念，提出一种更公正的清洁分担方案，以解决室友间因清洁习惯差异而产生的冲突，最大化共同生活的效益。

⚖️ 室友清洁分担困境：共同生活时，清洁习惯的差异是常见矛盾来源。一方可能认为室友邋遢，自己不得不承担更多清洁工作；另一方则可能觉得室友对清洁标准过高，自己被迫频繁打扫，双方都感觉承担了比独自生活时更多的清洁负担。

📊 传统分担方式的不公：文章分析了两种常见的室友清洁分担方式。一种是“邋遢鬼”对日益增长的脏乱无所谓，而“洁癖者”需要更频繁地清洁；另一种是“洁癖者”说服“邋遢鬼”按照自己的标准清洁。这两种方式都存在不公平，导致一方付出更多。

💡 Shapley值：公平解决方案的理论基础。作者提出应采用Shapley值来解决清洁分担问题。Shapley值是一种合作博弈论中的概念，用于公平地分配合作产生的收益或成本。在此情境下，它能帮助计算出每个室友应承担的清洁工作量，以达到最大程度的公平。

🧮 Shapley值计算示例：以清洁频率为例，若“洁癖者”独自生活时清洁频率为“邋遢鬼”的n倍，那么共同生活时，“洁癖者”应承担3n-1/2的清洁工作，而“邋遢鬼”应承担n+1/2的清洁工作。例如，如果“洁癖者”独居时清洁频率是对方的3倍，那么共同生活时，前者应承担后者两倍的清洁工作量。

Published on August 13, 2025 1:59 AM GMT

Perhaps you've had this experience before: you generally like living with your roommate, except when it comes to the mess they leave around. They seem content to wallow in their filth, so you feel like you always have to pick up after them to maintain a sane level of cleanliness. Because there are two of you, you have to do twice the work than if you were living alone.

Or perhaps you've had this experience before: you generally like living with your roommate, but they have far higher living standards than you would be content with. Your place hardly seems to get messy before they demand you pick up your filth. You don't want to make your roommate mad, so you do pick up a little more, but if feels like you're cleaning twice as often as you would be if you were living alone.

A mess is an example of a common bad. The two roommates could split up and find their own places, but they're probably generating more good by living together than if either one of them eats the extra costs. And there are extra costs:

They cannot just clean at the average of their preferred mess levels, because the 'clean freak' would freak out. Instead, the two most common ways of splitting the costs are either:

The 'filthy pig' is apathetic towards the growing mess, and the 'clean freak' cleans twice as frequently. Notice how the 'filthy pig' cleans less often than if they were living alone, while the 'clean freak' cleans more often.The 'clean freak' convinces the 'filthy pig' to clean as often as they do. Notice how the 'filthy pig' cleans more often than if they were living alone, while the 'clean freak' cleans just as often.

Neither of these are fair. They should be using Shapley values. Suppose the cleanlier roommate would clean $n$ times as often as the messier roommate, were they living alone. We have 

$$\begin{array}{cc}\text{\# Cleans({clean freak})}& =n\\ \text{\# Cleans({filthy pig})}& =1\\ \text{\# Cleans({clean freak, filthy pig})}& =2n\end{array}$$

so, living together, the 'clean freak' should clean $\frac{3n-1}{2}$ times for every $\frac{n+1}{2}$ times the messier roommate cleans. For example, if the cleanlier roommate would clean thrice as often on their own, they should be cleaning twice as much as their roommate.

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