On (Un)Knots and Dynamics in Games

We extend Kohlberg and Mertens' (1986) structure theorem on the Nash correspondence to show that its graph is not only homeomorphic to the underlying space of games, but that the homeomorphism extends to the ambient space of games times strategies, thus implying the graph is unknotted. This has several consequences for dynamics whose rest points are Nash equilibria. In particular, a homotopy property is established that allows to determine whether Nash equilibria or components of Nash equilibria - based directly on their local geometry on the graph - may be stable under a wide class of dynamics