Luca Sodomaco: A vector bundle approach to Nash equilibria

Abstract:

Using vector bundle techniques, we study the locus of totally mixed Nash equilibria of an \(n\)-player game. This approach reveals intriguing similarities and differences among totally mixed Nash equilibria, eigenvectors, and singular tuples of higher-order tensors. 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 its one-codimensional part is irreducible, and we study its degree. At a boundary format, we prove that the Nash discriminant variety also contains a two-codimensional component, in particular it is not irreducible. Finally, unlike singular tuples of tensors, a generic \(n\)-player game with an 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 nonempty Nash equilibria scheme. We prove that the Nash resultant variety is irreducible and determine its codimension and degree. We explicitly describe equations for Nash discriminant and resultant varieties for specific payoff tensor formats. Explicit computational examples support our results. The talk is based on a joint work with Hirotachi Abo and Irem Portakal.