Mathematical and Unified Research of Social Choice Theory

该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。(i)在有两个个体和三个备选方案的情况下（或在自由三元组性质的假设下），不存在满足Pareto原则、无关备选方案独立性和无独裁者的社会福利函数（传递二元社会选择规则）的Arrow不可能定理等价于二维球（圆）上的Brouwer不动点定理。(ii)二元社会选择规则的阿罗不可能性定理（Arrow impossibility theorem）指出不存在满足传递性帕累托原则、不相关替代项独立性（IIA）和无独裁者的二元社会选择规则，而Amartya Sen的自由悖论（liberal paradox）指出不存在满足非循环性、帕累托原则和最小自由主义的二元社会选择规则，二者在拓扑上是等价的。( 关于我们 2)HEX博弈定理和社会选择理论定理。(i)在对HEX博弈的标记规则进行一定的假设下，HEX博弈定理等价于社会选择理论中的Arrow不可能定理，即不存在满足传递性Pareto原则、无关方案独立性和无独裁者的二元社会选择规则。(ii)在对HEX博弈的标记规则作一定的假设下，HEX博弈定理等价于Duggan-Schwartz的防策略社会选择对应定理，即不存在满足防策略性、非强加性、剩余决断性和无独裁者条件的社会选择对应。(3)社会选择规则的可计算性。(i)如果一个社会选择函数存在一个独裁者，那么它在第二类可计算性的意义上是可计算的，但如果不存在独裁者，它就不是可计算的。(ii)无限社会中满足Pareto原理和不相关方案独立性（IIA）的传递性二元社会选择规则是否有独裁者是一个无法解决的问题，即不存在一个理想的计算机程序来判定满足Pareto原理和IIA的传递性二元社会选择规则是否有独裁者。这相当于停机问题的不可解性。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