On the Indices of Zeros of Nash Fields

This paper shows a fundamental property of vector fields representing dynamics on spaces of mixed strategies of normal form games whose zeros coincide with the Nash equilibria of the underlying games. The property shown is that the indices of components of zeros of any vector field in this class coincide with the local degrees of the projection map, mapping from the graph of the Nash equilibrium correspondence onto the space of games, evaluated at the corresponding components of Nash equilibria. This property is important since it implies that, for a large class of dynamics, the indices of components of zeros are completely determined by the geometry of the Nash equilibrium correspondence, thus providing a further link between evolutionary game theory, the theory of equilibrium refinements, and the geometry of Nash equilibrium