Suppose two players wish to divide a finite set of indivisible items, over which each distributes a specified number of points. Assuming the utility of a player’s bundle is the sum of the points it assigns to the items it contains, we analyze what divisions are fair. We show that if there is an envy-free (EF) allocation of the items, two other desirable properties—Pareto-optimality (PO) and Maximinality (MM)—can also be satisfied, rendering these three properties compatible. But there may be no EF division, in which case some division must satisfy a modification of Bentham’s (1789/2017) “greatest satisfaction of the greatest number” property, called maximum Nash welfare (MNW), that satisfies PO. However, an MNF division may be neither MM nor EFX, which is a weaker form of EF. We conjecture that there is always an EFX allocation that satisfies MM, ensuring that an allocation is maximin, precisely the property that Rawls (1971/1999) championed. We discuss four broader philosophical implications of our more technical analysis.

中文翻译：

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不可分割物品的两人公平划分——边沁与罗尔斯论嫉妒

假设两个玩家希望划分一组有限的不可分割的项目，每个人都分配指定数量的分数。假设玩家捆绑包的效用是其分配给其包含的物品的点数的总和，我们分析哪些划分是公平的。我们证明，如果项目存在无嫉妒（EF）分配，那么另外两个理想的属性——帕累托最优（PO）和极大值（MM）——也可以得到满足，从而使这三个属性兼容。但可能不存在 EF 除法，在这种情况下，某些除法必须满足边沁 (1789/2017)“最大数量的最大满足”属性的修改，称为最大纳什福利 (MNW)，满足 PO。然而，MNF 部门可能既不是 MM，也不是 EFX，EFX 是 EF 的较弱形式。我们推测总有一个 EFX 分配满足 MM，确保分配最大化，这正是罗尔斯（Rawls，1971/1999）所倡导的属性。我们讨论了更技术性分析的四个更广泛的哲学含义。