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Complete characterization of functions satisfying the conditions of Arrow’s theorem

机译：满足Arrow定理条件的函数的完整表征

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摘要

Arrow’s theorem implies that a social welfare function satisfying Transitivity, the Weak Pareto Principle (Unanimity), and Independence of Irrelevant Alternatives (IIA) must be dictatorial. When non-strict preferences are also allowed, a dictatorial social welfare function is defined as a function for which there exists a single voter whose strict preferences are followed. This definition allows for many different dictatorial functions, since non-strict preferences of the dictator are not necessarily followed. In particular, we construct examples of dictatorial functions which do not satisfy Transitivity and IIA. Thus Arrow’s theorem, in the case of non-strict preferences, does not provide a complete characterization of all social welfare functions satisfying Transitivity, the Weak Pareto Principle, and IIA. The main results of this article provide such a characterization for Arrow’s theorem, as well as for follow up results by Wilson. In particular, we strengthen Arrow’s and Wilson’s result by giving an exact if and only if condition for a function to satisfy Transitivity and IIA (and the Weak Pareto Principle). Additionally, we derive formulae for the number of functions satisfying these conditions.
机译：阿罗定理表明，满足传递性，弱帕累托原理（统一性）和无关选择的独立性（IIA）的社会福利功能必须是独裁的。当还允许非严格的偏好时，独裁的社会福利功能被定义为存在一个遵循严格的偏好的选民的功能。该定义允许许多不同的独裁功能，因为不一定遵循独裁者的非严格偏好。特别是，我们构造了不满足传递性和IIA的独裁功能的示例。因此，在非严格优先选择的情况下，阿罗定理不能完全满足满足及物性，弱帕累托原理和IIA的所有社会福利功能。本文的主要结果对Arrow定理以及威尔逊的后续结果进行了描述。尤其是，我们通过提供精确的条件（仅当且仅当条件满足函数的及物性和IIA（以及弱帕累托原理））来增强Arrow和Wilson的结果。此外，我们得出满足这些条件的函数数量的公式。

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