Some Preliminary Results

Abstract

First some preliminaries from matrix theory are reconsidered with a special focus on the Moore-Penrose matrix. Then results related to generalized conjugation theory from convex analysis are alluded which give a dual representation for n-person cooperative games with transferable utility. The dual representation of the characteristic function is denoted as an indirect function. This function is the generalized Fenchel transform of the characteristic function that contains the same information as games. Relying on the indirect function the pre-kernel of a TU game can be attained by an over-determined system of non-linear equations. This over-determined system of non-linear equations is equivalent to a minimization problem whose set of global minima coalesces with the pre-kernel set. The resultant objective function is a non-linear and non-convex function from which a member of the pre-kernel can be singled out through a modified Steepest Descent Method (MSDM). On the domain of this objective function equivalence relation can be identified allowing a partition of the payoff space.