On finite projective games

1. Preliminaries on simple games. Let N={ 1, 2, * * , n} be a finite set of n elements termed players. Let 9t be the class of all subsets S of N; the elements S of 91 are termed coalitions. If SC 1, let S+ denote the class of all supersets of elements of 8, and S* the class of all complements of elements of S; in symbols, g+ = [XcE j XDS for some SES], 8*= [XEmj N-XeS]. By a simple game is meant an ordered pair G = (N, sW) where W C 9t satisfies (1) W = W+, (2) WW * =0. The elements of W are termed winning coalitions. The elements of S = _-W are termed losing coalitions. The elements of (B = ?Gn ?* are termed blocking coalitions. A simple game2 is termed strong if 63 = 0. A simple game may be defined by specifying the class CWmC?W of minimal winning coalitions. By an imputation is meant an ordered n-tuple of real numbers x =(xi, x2, * * *, xn) such that3 xi>O and D=1 xi=1. If cUC91, let UO =91-(cq+)*; cUO is the class of all coalitions which intersect every element of m. If cU=Wm then CILO= S*=cW=UJ3. Suppose given a simple game (N, W), a nonempty class cLC'W, and real numbers a1, a2, * *, an such that