A General Structure Theorem for the Nash Equilibrium Correspondence

We consider n--person normal form games where the strategy set of each player is a non--empty compact convex subset of a Euclidean space, and the payoff function of player i is continuous in joint strategies and continuously differentiable and concave in player i''s strategy. No further restrictions (such as multilinearity of the payoff functions or the requirement that the strategy sets be polyhedral) are imposed. We demonstrate that the graph of the Nash equilibrium correspondence on this domain is homeomorphic to the space of games. This result generalizes a well--known structure theorem in Kohlberg and Mertens (On the Strategic Stability of Equilibria, Econometrica, 54, 1003--1037, 1986). It is supplemented by an extension analogous to the unknottedness theorems in Demichelis and Germano (On (Un)knots and Dynamics in Games, Games and Economic Behavior, 41, 46--60, 2002): the graph of the Nash equilibrium correspondence is ambient isotopic to a trivial copy of the space of games.