Social Choice Scoring Functions

Abstract

Let a committee of voters be considering a finite set $A = \{ {a_1 ,a_2 , \cdots ,a_m } \}$ of alternatives for election. Each voter is assumed to rank the alternatives according to his preferences in a strict linear order. A social choice function is a rule which, to every finite committee of voters with specified preference orders, assigns a nonempty subset of A, interpreted as the set of “winners”. A social choice function is consistent if, whenever two disjoint committees meeting separately choose the same winner(s), then the committees meeting jointly choose precisely these winner(s). The function is symmetric if it does not depend on the names of the various voters and the various alternatives. It is shown that every symmetric, consistent social choice function is obtained (except for ties) in the following way: there is a sequence $s_1 ,s_2 , \cdots $, $s_m $ of m real numbers such that if every voter gives score $s_i $ to his ith most preferred alternative, then the alternative with highest score (summed over all voters) is the winner.