Generic Instability of Majority Rule

Abstract

Majority rule voting with smooth preferences on a smooth policy space W is examined. It is shown that there is an integer w ( n ), which is 2 when the size of the society n is odd and 3 when n is even such that when the dimension of W is at least w ( n ) then, for almost preference profiles on W , the core of the voting game is empty when the dimension of W exceeds w ( n ) then for almost all preference profiles on W , there exist dense preference cycles in W . Moreover in dimension w (n) + 1 the policy space can be partitioned into a finite number of path connected components, such that any two points in one of the components can be connected by a majority voting trajectory. In dimension greater than w (n) + 1 there exists only one such component.