Universality of Nash Equilibria

Every real algebraic variety is isomorphic to the set of totally mixed Nash equilibria of some three-person normal form noncooperative game with perfect information and rational payoffs, and also to the set of totally mixed Nash equilibria of an N-person game with rational payoffs in which each player has two pure strategies. From the Nash-Tognoli Theorem it immediately follows that every compact differentiable manifold is diffeomorphic to the set of totally mixed Nash equilibria of some game.