Balancedness,nonemptiness of the core and the existence of a majority equilibrium

Consider a society with a finite number,n, of individuals who have to choose an alternative from a setA in them-dimensional Euclidean Space, R ^m . Assuming that the preference relation overA of every individual is convex and continuous,Greenberg [1979] showed that if the set of winning coalitions (i.e. those that have the veto power) consists of all coalitions with more thanmn/(m+1) individuals, then there exists an alternativea ^* ?A which is a majority equilibrium, or equivalently, that the core of the induced game is nonempty. In the present paper we strengthen this result by showing that this game not only has a nonempty core but is also balanced.