An Observation related to the method of Lemke-Hobson

Introduction Even though Nash´s theorem asserts the existence of equilibrium points for the mixed extension of a normal game, it does not tell us how to find them. Even in the case of two matrices or two person games many algorithms have been proposed by Vorobev[13], Kuhn[3] and Mangasarian[5] to determine all equilibrium pairs, they are more of theoretical interest than for actual computation. The algorithm proposed by Lemke and Howson[4] seems until now to be one of the most effective for finding an equilibrium pair. In the case of n-person games there have been some attempts in order to get the computation of an equilibrium point, let us mention the work of Rosenmüller[11], Sobel[12], Wilson[14] and Garcia, Lemke and Luethi[2]. This last reference is relationed with the simplicial approximation of equilibrium points. This suggested the constructive, combinatorial approach of Scarf for finding, among other things, approximations to fixed-points of continuous mappings. However, the effective computation even in simple cases and low dimension is still open and very intricate questions which have to be attacked and solved in the future. This paper deals with a new approach based on the rather effective algorithm Lemke and Howson to general n-person games. The definition of general n-person games is going to be accordingly in the next pages.