A decomposition of the class of semiconvex games

The paper studies the classSC N of cooperative games with player setN which have the semiconvexity property.SC N is decomposed into an algebraic sum of convex cones of games for which generating sets are available. The union of these sets thus forms a generating set forSC N . Special attention is paid to one of the considered cones, in the decomposition ofSC N . In particular, the so called airport savings gamesW y, ℝ N , defined byw y (S)=Σ jεS Y i −max jεS Y j for 0 ≠S⋞N, are emphasized

Decomposition of Games with Non-empty Core into Veto-Controlled Simple Games

The non-negative games with non-empty core form a polyhedral cone. Extreme directions of this convex cone correspond to veto-controlled simple games. In this paper a constructive method is described to represent a non-negative game with non-empty core as a positive linear combination of veto-controlled simple games. As an application a constructive proof is suggested of a result of spinetto (see [5]) dealing with the extreme points of the compact and convex set of nonnegative (0, 1)-normalized games with non-empty core

On Linear Inequality Systems without Strongly Redundant Constraints

For an (m x n)-matrix A the set C(A) is studied containing the constraint vectors b of R(m) without strongly redundant inequalities in the system (Ax less-than-or-equal-to b, x greater-than-or-equal-to 0). C(A) is a polyhedral cone containing as a subset the cone Col(A) generated by the column vectors of A. This paper characterizes the matrices A for which the equality C(A) = Col(A) holds. Furthermore, the matrices A are characterized for which C(A) is generated by those constraint vectors beta(i) is-an-element-of C(A), i is-an-element-of {1, 2, ..., m}, for which the feasible region {x is-an-element-of R+n: Ax less-than-or-equal-to beta(i)} equals {x is-an-element-of R+n: (Ax)i less-than-or-equal-to 1}. Necessary conditions are formulated for a constraint vector to be an element of C(A). The class of matrices is characterized for which these conditions are also sufficient