Generic Instability of Majority Rule

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 (i) when the dimension of W is at least win) then, for almost preference profiles on W, the core of the voting game is empty (ii) when the dimension of W exceeds win I then for almost all preference profiles on W, there exist dense preference cycles in W. Moreover in dimension w(n) + I 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. 1.