投票悖论概率计算的基本定理  

The fundamental theorem for the probability calculations of the paradox of voting

作　　者：胡毓达 HU Yuda(School of Mathematical Sciences,Shanghai Jiaotong University,Shanghai 200240,China)

机构地区：[1]上海交通大学数学科学学院,上海200240

出　　处：《运筹学学报》2020年第2期1-13,共13页Operations Research Transactions

摘　　要：多数偏好规则是求解群体最优化问题最重要和应用最广泛的规则之一.但是,在使用这一规则对给定的群体最优化问题寻求最优解中,有时会发生群体偏好循环排序的"投票悖论"现象,从而发生"投票悖论"现象的概率计算,便成为群体最优化中一个基本的研究课题.为此,引进群体在方案集上的"投票悖论排序剖面"和"投票悖论选型剖面"的概念.借助于这两个概念,建立了群体中每一个体对所有方案的偏好排序都可各具不同概率分布的一般情况下,群体对问题进行方案择优时,发生"投票悖论"概率计算的基本定理.于是,解决了求解群体最优化时发生"投票悖论"的概率计算研究中,长期未能得到彻底解决的基本问题.The majority preference rule is one of the most important and widely used rule in solving group optimization problems.However,in the process of using this rule to find the optimal solution for a given group optimization problem,sometimes there will be a“paradox of voting”phenomenon of group preference cyclic ordering.Therefore,the probability calculations of“paradox of voting”has become a fundamental research topic in group optimization.In this paper,the concepts of“sorting profile of the paradox of voting”and"selection profile of the paradox of voting"of a group on scheme set are introduced.With the help of them,a fundamental theorem for the probability calculations of the“paradox of voting”is established under the general condition that each individual in a group has different probability distribution for the preference ranking of all schemes,and when a group chooses the best scheme for a problem.Thus,the fundamental problem that has not been solved thoroughly for a long time in the study for the probability calculations of the“paradox of voting”is solved.

关 键 词：群体最优化 多数偏好规则 投票悖论 概率 

分 类 号：O221[理学—运筹学与控制论]