A vector bundle approach to Nash equilibria

Using vector bundle techniques, we study the locus of totally mixed Nash equilibria of an n-player game in normal form.

When the payoff tensor format is balanced, we define and study the Nash discriminant variety, that is the algebraic variety of n-player games whose Nash equilibria scheme is either non-reduced or has a positive dimensional component. We verify that it has codimension one and compute its degree in some formats. In the format 2x2x2, we describe all singular strata of the Nash discriminant variety.

At a boundary format, we prove that the Nash discriminant variety also contains a two-codimensional component, in particular it is not irreducible.

Finally, a generic n-player game with unbalanced payoff tensor format does not admit totally mixed Nash equilibria. We define the Nash resultant variety as the proper subvariety of games admitting a positive number of totally mixed Nash equilibria. We prove that the Nash resultant variety is irreducible and determine its codimension and degree.