Competitive equilibrium (CE) is a fundamental concept in market economics. Its efficiency and fairness properties make it particularly appealing as a rule for fair allocation of resources among agents with possibly different entitlements. However, when the resources are indivisible, a CE might not exist even when there is one resource and two agents with equal incomes. Recently, Babaioff and Nisan and Talgam-Cohen (2017–2019) have suggested to consider the entire space of possible incomes, and check whether there exists a CE for almost all income-vectors—all income-space except a subset of measure zero. They proved various existence and non-existence results, but left open the cases of four goods and three or four agents with monotonically-increasing preferences. This paper proves non-existence in both these cases, thus completing the characterization of CE existence for almost all incomes in the domain of monotonically increasing preferences. Additionally, the paper provides a complete characterization of CE existence in the domain of monotonically decreasing preferences, corresponding to allocation of chores. On the positive side, the paper proves that CE exists for almost all incomes when there are four goods and three agents with additive preferences. The proof uses a new tool for describing a CE, as a subgame-perfect equilibrium of a specific sequential game. The same tool also enables substantially simpler proofs to the cases already proved by Babaioff et al. Additionally, this paper proves several strong fairness properties that are satisfied by any CE allocation, illustrating its usefulness for fair allocation among agents with different entitlements.

中文翻译：

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几乎所有收入的竞争均衡：存在与公平

竞争均衡（CE）是市场经济学中的基本概念。它的效率和公平性使其在具有不同权利的代理商之间公平分配资源的规则特别有吸引力。但是，当资源不可分割时，即使有一种资源和两个收入相同的代理人，CE也可能不存在。最近，Babaioff和Nisan和Talgam-Cohen（2017-2019）建议考虑可能收入的整个空间，并检查几乎所有收入向量是否都存在CE—除度量零的子集外的所有收入空间。他们证明了各种存在和不存在的结果，但没有公开四个具有单调递增偏好的商品和三个或四个代理商的案例。本文证明了这两种情况都不存在，从而完成了在单调递增的偏好范围内几乎所有收入的CE存在的表征。另外，本文提供了CE在单调递减的偏好域中的存在的完整特征，对应于琐事的分配。从积极的一面来看，本文证明了当有四个具有附加偏好的商品和三个代理商时，CE几乎存在于所有收入中。证明使用一种新的工具来描述CE，作为特定顺序游戏的子游戏完美平衡。相同的工具还可以对Babaioff等人已经证明的情况进行简单得多的证明。此外，本文证明了任何CE分配都可以满足的几个强大的公平性，从而说明了其对于具有不同权利的代理之间公平分配的有用性。