Social Choice Function

看到关于Web服务质量选择的决策算法，提到一种多属性的群决策服务选择算法
开篇引入了一个SOCIAL CHOICE FUNCTION，并以此为基础进行计算
仔细看看到底是个啥东东？
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早在上世纪四十年代末、五十年代初，现代经济学巨匠之一的阿罗教授（K.Arrow），就在其博士论文中证明了许多《信报》读者可能耳熟能详的「不可能性定理」（Arrow Impossibility Theorem）。阿罗教授首次用严谨的数字推算，论证不可能存在任何一个社会选择函数（social choice function），能同时满足数条看上去相当合情合理的数学定理（axiom），如个人的偏好具有传递性（亦即喜欢Ａ超过Ｂ而且喜欢Ｂ超过Ｃ的人，应该喜欢Ａ超过Ｃ）、帕累托原则、不相关选择之间的独立性、不能有强制或独裁等。换句话说，想把所有个人偏好整合成一个社会偏好，上述这些公理中至少有一条不能被满足。
尽管这个超前的博士论文害得阿罗教授毕业后找不到正式工作，但这篇论文却开创了一个划时代的学科，「社会选择理论」（social choice theory），今天还有专门的学会。
较少为人注意的是七十年代初，一位哲学家（Alan Gibbard）和两位数理经济学（Mark Satterthwaite与Prasanta K. Pattanaik），几乎同时独立发现并证明，不可能存在任何一种选举制度（voting）能免于被操纵（manipulation）。这就是着名的Gibbard-Satterthwaite定理。其分析方法类似于阿罗不可能性定理，也是从数条看似相当合情合理的数学公理出发，结果推算出论来。这方面的研究在上世纪七、八十年代成果辈出。
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If there are only two candidates, the answer is clear--- choose the one who would win the most votes in a head-to-head election. But with three or more candidates, when each voter has ranked his or her candidate preferences, the answer is less obvious.
Mathematically we can formalize the question in this way. A social choice function is a function that takes lists of people‘s ranked preferences and outputs a single alternative (the "winner" of the election). So the question becomes: is there a "good" social choice function that represents "the will of the people"?
Consider the following situation with 3 voters and 3 candidates:
Suppose Voter 1 prefers A to B to C.
Suppose Voter 2 prefers B to C to A.
Suppose Voter 3 prefers C to A to B.
Notice that no matter who is selected as the "social choice" for this set of lists, then 2/3 of the voters will be "unhappy" in the sense that those voters prefer another candidate to the one chosen by the social choice function! (For instance, if A is chosen as the winner, then Voters 2 and 3 will prefer C to A.)
This paradox, due to Maurice de Condorcet in 1785, shows that it is not always possible for a social choice function to pick a candidate that will beat all other candidates in pairwise comparisons. If there is a candidate that does, then that candidate is called a Condorcet winner.
The Math Behind the Fact:
The study of social choice functions and related questions is called social choice theory. There are other famous impossibility results: most notablyArrow‘s Impossibility Theorem.
参考上述文献，可以得出，SOCIAL CHOICE FUCNTION是一个这样的函数：输入为若干决策人对多个被选择对象的排序结果，输出则为综合了所有决策人排序结果后给出的最优先被选择的唯一确定的对象。