Abstract
We use vector bundles to study the locus of totally mixed Nash equilibria of
an $n$-player game in normal form, which we call Nash equilibrium scheme. When
the payoff tensor format is balanced, we study the Nash discriminant variety,
i.e., the algebraic variety of games whose Nash equilibrium scheme is
nonreduced or has a positive dimensional component. We prove that this variety
has codimension one. We classify all components of the Nash equilibrium scheme
of binary three-player games. We prove that if the payoff tensor is of boundary
format, then the Nash discriminant variety has two components: an irreducible
hypersurface and a larger-codimensional component. A generic game with an
unbalanced payoff tensor format does not admit totally mixed Nash equilibria.
We define the Nash resultant variety of games admitting a positive number of
totally mixed Nash equilibria. We prove that it is irreducible and determine
its codimension and degree.