On equilibria in finite games

Abstract

We are concerned with Nash equilibrium points for n-person games. It is proved that, given any real algebraic number α, there exists a 3-person game with rational data which has a unique equilibrium point and α is the equilibrium payoff for some player. We also present a method which allows us to reduce an arbitrary n-person game to a 3-person one, so that a number of questions about general n-person games can be reduced to consideration of the special 3-person case. Finally, a completely mixed game, where the equilibrium set is a manifold of dimension one, is constructed. © 1979 Physica-Verlag.