Nash Loci

A classical problem in game theory is that of determining a Nash equilibrium for a given finite-player game. In this talk we will turn the problem around: When designing a game we might want to ensure that the game is in some sense “fair” or “well-balanced". One criterion for fairness could be that there exists a Nash equilibrium in which all players play the same mixed strategy. A criterion for well-balancedness could be that there exists a Nash equilibrium where each player plays all of his strategies with equal probability. This motivates us to study “Nash loci”, i.e. the algebraic loci of all games which admit a Nash equilibrium that satisfies a set of prescribed polynomial equations.

In the talk we will define Nash loci carefully and illustrate them in several examples. We will further determine their dimension and degree in general and their equations for two-player games. The special structure of the defining equations for Nash equilibria will allow us to relate Nash loci to “multigrade associated varieties”, which live in a product of Grassmannians and generalise the notion of Cayley-Chow hypersurfaces. This will lead us to a natural stratification of Nash loci.

This talk is based on the following preprint with Luca Sodomaco: arxiv.org/abs/2608.27300