Nashtopes and their A-discriminants

Abstract

An $n$-player game in normal form is said to be of format $\boldsymbol{d} = (d_1, \dots, d_n)$ if the $i$-th player has $d_i$ options for $1 \le i \le n$, and is said to be balanced if $d_n - 1 \le \sum_{i=1}^{n-1}(d_i - 1)$. The Nash equilibria of the game correspond to solutions of the system of indifference equations for the game, with payoff differences as coefficients and strategy probabilities as variables. As such, the mixed discriminant of this system gives some information about which games could have an unusual number of totally mixed Nash equilibria. This mixed discriminant can be regarded as an $\mathcal{A}$-discriminant for some matrix $\mathcal{A}$ in the sense of [Gelfand--Kapranov--Zelevinsky 1994], and the convex hull of the columns of $\mathcal{A}$ is a polytope, which we call the Nashtope. We study this $\mathcal{A}$-discriminant for each balanced game format with at least three players. This discriminant is a subset of the Nash discriminant variety introduced in [Abo--Portakal--Sodomaco 2026]. We compute the codimension and degree of this discriminant for all balanced game formats, including on-boundary formats, using the Euler obstruction formula in [Matsui--Takeuchi 2011]. Our $\mathcal{A}$-discriminant locus corresponds to a special multigraded Hurwitz locus studied in [Pratt--Sodomaco--Sturmfels 2026]. As a consequence of [Pokidkin 2026], we obtain the codimensions and degrees of the irreducible components of the Nash discriminant variety for each balanced game format with at least three players, fully resolving Problem 3.8 of [Abo--Portakal--Sodomaco 2026].