What majority decisions are possible

Abstract

The main result is the following: Let X be a finite set and D be a non empty family of choice functions for (X choose 2) closed under permutation of X. Then the following conditions are equivalent: (A) for any choice function c on (X choose 2) we can find a finite set J and c_j in D for j in J such that for any x not= y in X : c{x,y}=y Leftrightarrow |J|/2<| {j in J:c_j{x,y}= y}| (so equality never occurs) (B) for some c in D and x in X we have |{y: c{x,y}=y}| not= (|X|-1)/2 . We then describe what is the closure of a set of choice functions by majority; in fact, there are just two possibilities (in section 3). In section 4 we discuss a generalization.