Mathematical and Unified Research of Social Choice Theory

本项目的主要成果如下：(1)代数拓扑在社会选择理论中的应用。(1)在有两个个体和三个选择的情况下（或在自由三重性质的假设下），不存在满足帕累托原则、不相关选择的独立性、没有独裁者的社会福利函数（传递二元社会选择规则）的阿罗不可能定理等价于二维球（圆）上的布劳威尔不动点定理。(2)二元社会选择规则的阿罗不可能定理，即不存在满足传递性帕累托原则、无关选择独立性（IIA）和不存在独裁者的二元社会选择规则；以及阿马蒂亚·森二元社会选择规则的自由主义悖论，即不存在满足不循环性、帕累托原则和最小自由主义的拓扑等价。（…More 2） HEX博弈定理和社会选择理论定理。(i)在关于HEX博弈标记规则的某些假设下，HEX博弈定理等价于社会选择理论中的阿罗不可能性定理，即不存在满足传递性帕累托原则、不相关选项独立、不存在独裁者的二元社会选择规则。（ii）在对HEX博弈标记规则的某些假设下，HEX博弈定理等价于无策略社会选择对应的Duggan-Schwartz定理，即不存在满足无策略性、非强制性、剩余决定论和无独裁者的社会选择对应。(3)社会选择规则可计算性。(i)如果存在一个社会选择函数的独裁者，它在第二类可计算意义上是可计算的，但如果不存在独裁者，它是不可计算的。（ii）在无限社会中满足帕累托原则和无关选择独立性（IIA）的传递二元社会选择规则是否有独裁者是一个无法解决的问题，即对于满足帕累托原则和IIA的传递二元社会选择规则是否有独裁者的理想计算机程序是不存在的。它等价于停止问题的不可解性。少

Principal achievements of this project are as follows.(1) Applications of algebraic topology to social choice theory.(i) In the case where there are two individuals and three alternatives (or under the assumption of free-triple property) the Arrow impossibility theorem for social welfare functions that there exists no social welfare function (transitive binary social choice rule) which satisfies Pareto principle, independence of irrelevant alternatives, and has no dictator is equivalent to the Brouwer fixed point theorem on a 2-dimensional ball (circle).(ii) Arrow impossibility theorem for binary social choice rules that there exists no binary social choice rule which satisfies transitivity Pareto principle, independence of irrelevant alternatives (IIA), and has no dictator, and Amartya Sen's liberal paradox for binary social choice rules that there exists no binary social choice rule which satisfies acyclicity, Pareto principle and the minimal liberalism are topologically equivalent.( … More 2) HEX game theorem and theorems of social choice theory.(i) Under some assumptions about marking rules of HEX games, the HEX game theorem is equivalent to the Arrow impossibility theorem of social choice theory that there exists no binary social choice rule which satisfies transitivity Pareto principle, independence of irrelevant alternatives and has no dictator.(ii) Under some assumptions about marking rules of HEX games, the HEX game theorem is equivalent to the Duggan-Schwartz theorem for strategy-proof social choice correspondences that there exists no social choice correspondence which satisfies the conditions of strategy-proofness, non-imposition, residual resoluteness, and has no dictator.(3) Computability of social choice rules.(i) If there exists a dictator for a social choice function, it is computable in the sense of Type two computability, but if there exists no dictator it is not computable.(ii) The problem whether a transitive binary social choice rule satisfying Pareto principle and independence of irrelevant alternative (IIA) has a dictator or has no dictator in an infinite society is an unsolvable problem, that is, there exists no ideal computer program for a transitive binary social choice rule satisfying Pareto principle and IIA that decides whether the binary social choice rule has a dictator or has no dictator. And it is equivalent to unsolvability of the halting problem. Less