经济学中的相交集族与锥拓扑
Intersecting families of sets and the topology of cones in economics
摘要：
经济学中的两个经典问题，市场均衡的存在性和社会选择函数的存在性，在这里通过与经济相关的一族圆锥的性质来形式化。最近建立了求解前者的一个充要条件是锥族的非空交，求解后者的一个这样的条件是其子族的并的非圆性。我们显示出一个意想不到的，但明确的两个问题之间的联系，通过建立一个对偶性质的同调群的神经定义的家庭锥。特别是，我们证明了锥族的交集是非空的当且仅当每个子族都有非循环工会，从而确定了解决这两个经济问题的两个条件。除了他们在经济学中的应用，结果显示显着扩展几个经典定理，提供统一和简单的证明：Helly定理，Caratheodory的表示定理，Knaster-Kuratowski-Marzukiewicz定理，Brouwer的不动点定理，和Leray定理的非循环覆盖。
英文摘要：
Two classical problems in economics, the existence of a market equilibrium and the existence of social choice functions, are formalized here by the properties of a family of cones associated with the economy. It was recently established that a necessary and sufficient condition for solving the former is the nonempty intersection of the family of cones, and one such condition for solving the latter is the acyclicity of the unions of its subfamilies. We show an unexpected but clear connection between the two problems by establishing a duality property of the homology groups of the nerve defined by the family of cones. In particular, we prove that the intersection of the family of cones is nonempty if and only if every subfamily has acyclic unions, thus identifying the two conditions that solve the two economic problems. In addition to their applications to economics, the results are shown to extend significantly several classical theorems, providing unified and simple proofs: Helly's theorem, Caratheodory's representation theorem, the Knaster-Kuratowski-Marzukiewicz theorem, Brouwer's fixed point theorem, and Leray's theorem on acyclic covers.
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