关于投票循环的条件及多目标决策

On the Conditions of Voting Circular and Multiobjective Decision

设有 n个方案 x1,x2 ,… ,xn,有 n组人对这 n个方案进行过半数投票。令人感兴趣的是 ,何时会出现如下的投票循环问题 ,即 x1以 ( n- 1 )∶ 1击败 x2 ;x2 以 ( n- 1 )∶ 1击败 x3,… ;xn-1以 ( n- 1 )∶ 1击败xn;而 xn 却以 ( n- 1 )∶ 1击败 x1。文中经过数学证明 ,得到了该投票循环的一个充分必要条件 。

Let x 1,x 2,…,x n be n programs for voting. There are n groups of people to vote on these programs by using the morethan half voting rule. What iterest us is when the following voting circular problem occurs:x 1 defeating x 2 by (n-1)∶1, x 2 defeating x 3 by (n-1)∶1,…,x n-1 defeating x n by (n-1)∶1, while x n defeating x 1 by (n-1)∶1. Through rigid mathematical analysis, we get the sufficient and necessary condition of this voting circular, which is used in multiobjective decision.