Slicing the Nash equilibrium manifold

This paper uses tools on the structure of the Nash equilibrium correspondence of strategic-form games to characterize a class of fixed-point correspondences, that is, correspondences assigning, for a given parametrized function, the fixed-points associated with each value of the parameter. After generalizing recent results from the game-theoretic literature, we deduce that every fixed-point correspondence associated with a semi-algebraic function is the projection of a Nash equilibrium correspondence, and hence its graph is a slice of a projection, as well as a projection of a slice, of a manifold that is homeomorphic, even isotopic, to a Euclidean space. As a result, we derive an illustrative proof of Browder’s theorem for fixed-point correspondences.