PaperlayerView exact revision on arXiv

Source-linked AI summary

Nashtopes and their A-discriminants

Elke Neuhaus, Irem Portakal, Luca Sodomaco, Katherine Tung

arXiv:2610.29889v1math.COcs.GTmath.AG

TL;DR

The dimensions and degrees of the Nash discriminant variety were not fully known for balanced formats, despite its role in describing nongeneric equilibrium behavior. The paper studies the associated A-discriminant and uses its degree formulas to determine the irreducible components’ codimensions and degrees. It establishes irreducibility for strictly balanced formats and identifies two components on the boundary.

  • Problem

    For balanced and on-boundary formats, the dimension and degree of the Nash discriminant variety were not completely known.

    Source: §1 · p. 2

  • Method

    The paper studies the A-discriminant as a multigraded Hurwitz locus and applies its degree formulas with results of [Pok26] to determine Nash discriminant components.

    Source: §1 · pp. 2–3

  • Results

    The Nash discriminant variety is irreducible for strictly balanced formats, while on the boundary it has two components of codimensions 1 and 2.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The degree formula provides a quantitative geography of degenerate loci, with format dependence describing their distribution across game space.

    Source: §1 · p. 3

    The analysis assumes all systems A are square, and on-boundary degree calculations require a correction term.

    Source: §3.6 · p. 22, §5 · p. 32

Abstract

from arXiv · show

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].

1. Introduction

The paper studies the A-discriminant of Nash equilibrium systems to characterize degenerate behavior across balanced game formats. It computes component codimensions and degrees for the Nash discriminant variety, addressing previously unresolved cases.

1. Introduction

  • The generic number of totally mixed equilibria is constant on a Zariski-open set of games and equals the mixed-volume count.This generic count follows from the Bernstein-Khovanskii-Kushnirenko theorem.Source: §1 · p. 2
  • The paper resolves the previously open dimension and degree questions for the Nash discriminant variety in balanced and on-boundary formats.For balanced formats, the variety contains a real-codimension-one hypersurface component; the remaining dimension and degree information was incomplete.Source: §1 · pp. 2–3
  • For each game format, the Nashmatrix is the Cayley matrix of the Nash-equilibrium equations’ monomial exponents, and its columns’ convex hull is the Nashtope.The paper computes the dimension and degree of the resulting A-discriminant for balanced formats.Source: §1 · p. 2
  • For strictly balanced formats, the Nash discriminant variety is irreducible; on the boundary, it has components of codimensions 1 and 2.The boundary components’ degrees are determined by the degree of the A-discriminant and the result in [APS26, Theorem 3.26].Source: §1 · p. 2
  • The degree formula gives a quantitative account of how degenerate behavior varies with format, while the link to actual equilibria in the open simplex is not treated.This question is distinct from the complex-algebraic study of degeneracies in the Nash equilibrium scheme.Source: §1 · p. 3
  • The A-discriminant is a multigraded Hurwitz locus, and the paper attains the known upper bound for its multidegree in strictly balanced formats using polynodality.The relevant geometric property is established for the mixed Segre embedding of the multiprojective strategy space.Source: §1 · p. 3
  • The authors provide Macaulay2 and Julia software for generating finite-player games on the A-discriminant variety from an input Nashmatrix.The code and documentation are available in the Zenodo repository.Source: §1 · p. 3

2. Preliminaries

The preliminaries set up the Nash equilibrium scheme and its associated matrix and polytope, then identify the A-discriminant as a locus of interest within the Nash discriminant. A four-player binary example illustrates the potentially complex algebraic behavior captured by this construction.

2. Preliminaries

  • For balanced formats, the Nash equilibrium scheme is generically zero-dimensional, while unbalanced formats have generically empty schemes.For balanced games, the generic degree is given by the mixed volume of the Newton polytopes.Source: §2 · p. 4
  • The Nashmatrix is a Cayley matrix built from exponent vectors of monomials in the Nash equilibrium equations, and its column convex hull is the Nashtope.The columns are vertices of the Nashtope, whose faces are called Nashfaces.Source: §2 · pp. 4–5
  • The A-discriminant records coefficient vectors for which the associated polynomial hypersurface H_A,z is singular.The polynomial is formed from the columns of A and coefficient vector z.Source: §2 · p. 5
  • The A-discriminant variety ∇A is a subset of the Nash discriminant variety ∆(d), and its dimension and degree are the section’s central objects of study.It is initially unclear whether ∇A is an irreducible component or the entire variety.Source: §2 · p. 5
  • For format (2, 2, 2, 2), the Nashmatrix has size 12 × 32 and the A-discriminant is a degree-80 hypersurface in P^31.The defining polynomial involves 32 unknown coefficients.Source: §2 · p. 5
  • A sampled point on the format-(2, 2, 2, 2) A-discriminant yields a degree-9 nonreduced equilibrium scheme supported at 8 points, with the totally mixed equilibrium having multiplicity two.The scheme is zero-dimensional.Source: §2 · p. 6

3. The degree of the A-discriminant for balanced game formats

The section computes the degree and codimension of the A-discriminant for balanced game formats, using the combinatorial structure of balanced Nashtopes. It develops face-dimension and Euler-obstruction calculations with normalized volumes to give degree formulas for strictly balanced and on-boundary formats, including a closed formula for binary games and an application to format (2, 2, 3).

3. The degree of the A-discriminant for balanced game formats

  • The A-discriminant has codimension 1 for strictly balanced formats and codimension 2 for on-boundary formats.The section aims to compute its degree and also obtains its dimension and the combinatorial structure of the Nashfaces.Source: §3 · p. 6

3.1. Roadmap for the degree calculation

  • The degree calculation applies a formula for general dual toric varieties using Nashtope face volumes and Euler obstructions; the first nonzero δ_i gives the codimension and degree.Specifically, r = min{i | δ_i ≠ 0} and deg(∇A) = δ_r.Source: §3.1 · p. 7
  • The Euler obstruction depends on the lattice generated by A’s columns, and the calculation develops the auxiliary functions u and i and normalized volumes.These quantities are then used to calculate the Euler obstruction and the coefficients δ_i.Source: §3.1 · pp. 7–8

3.2. The faces of balanced Nashtopes

  • Balanced Nashtope faces are indexed by auxiliary face strings, with prismatic strings corresponding to products of simplices.The strings include prismatic and nonprismatic types.Source: §3.2 · pp. 8–9
  • The map from face strings to Nashtope faces is a bijection.Each string maps to a face, and every face is represented by a string.Source: §3.2 · pp. 10–11
  • For any Nashtope face, its vertices within the specified product-of-simplices face form the vertex set of an i-prismatic face string, when that intersection is nonempty.This characterizes how such intersections relate to the prismatic strings used in the face indexing.Source: §3.2 · p. 10
  • A face string and its image under the face-string map have the same dimension.This equality supports using strings to describe the face structure of the Nashtope.Source: §3.2 · p. 11

3.2. The faces of balanced Nashtopes.

  • Under either stated removal condition, a nonprismatic face string covering a face of dimension d has image dimension d + 1.The conditions remove an element from a C_i while at least two C_i remain nonempty, or from a D_i while all D_i remain nonempty.Source: §3.2 · p. 12
  • In the singleton configuration with only two nonempty singleton C_i, a nonprismatic face covering a 0-dimensional prismatic face has image dimension 1.This case has all D_i singleton and the prismatic face is formed by replacing one D_i with its C_i.Source: §3.2 · p. 12

3.3. The functions u and i for Nashfaces.

  • For a prismatic Nashface, replacing one option in a D_j (j ≠ i) or C_i yields an affine basis for its generated semigroup.The basis is taken relative to the image of a vertex of the face.Source: §3.3 · p. 12
  • For a nonprismatic Nashface, an affine spanning set also includes cross-moves replacing δ_j with γ_j and γ_i with δ_i.With a fixed δ_i, the corresponding affine vector space has a basis described by the lemma.Source: §3.3 · p. 13
  • For nested nonprismatic Nashfaces, the normalized volume Vol_M(G)(K \ K+) equals 1, so u(F, F′) = 1.The lemma derives the unit volume from a linearly independent generating set in the quotient space.Source: §3.3 · pp. 14–15
  • For nested prismatic Nashfaces, u(F, F′) = 1.Source: §3.3 · p. 15
  • When the affine lattice generated by A has codimension 1 in the lattice generated by A, Vol_MA(K \ K+) equals the normalized volume of Q in the affine lattice.Here Q is the convex hull of A, and K and K+ are as defined for the generated semigroup.Source: §3.3 · p. 16
  • For nested Nashfaces F′ ≤ F, the index i(F, F′) always equals 1.This holds for every type of face of the Nashtope.Source: §3.3 · p. 17

3.4. The Euler obstruction function for Nashfaces.

  • Every nonprismatic Nashface has Euler obstruction 1.The Euler obstruction of an i-prismatic face equals that of its unique maximal prism containing it.Source: §3.4 · pp. 18, 20
  • For n-prismatic faces, the Euler-obstruction value differs between strictly balanced and on-boundary formats.The passage states that it is d_n for strictly balanced formats and gives a distinct value for on-boundary formats, but the latter expression is incomplete in the supplied text.Source: §3.4 · p. 20

3.5. The normalized volumes of Nashfaces.

  • A generalized function on sequences of boxed and unboxed integers is introduced to represent face volumes and aid degree computations.When the sequences correspond to face strings, the function gives their normalized volume.Source: §3.5 · p. 20

3.6. The degree of the A-discriminant for strictly balanced game formats.

  • The coefficient computation requires a correction term in the on-boundary case.The passage directs readers to the proof of Lemma 3.47 for comparison.Source: §3.6 · p. 22
  • For strictly balanced formats, δ1 is expressed as a coefficient of a formal power series in z1, …, zn.Theorem 3.43 gives a coefficient formulation after rewriting the degree computation in the z variables.Source: §3.6 · p. 23
  • For every strictly balanced game format, the A-discriminant is a hypersurface, with degree given by the coefficient specified in Corollary 3.44.The supplied passage does not include the full coefficient expression.Source: §3.6 · p. 24
  • For binary games, deg(∇A) = n [(n −2) · !n + (n −3) · !(n −1)], where !n counts derangements of [n].The result sharpens the degree bound in [PSS26, Example 5.6].Source: §3.6 · p. 24

3.7. The degree of the A-discriminant for on-boundary game formats.

  • For on-boundary formats, δ2 equals a coefficient of a formal power series: the coefficient of (x1y1)^(d1−1) ··· (xnyn)^(dn−1).The passage identifies this coefficient representation but does not include the series expression in the supplied text.Source: §3.7 · p. 26
  • For format (2, 2, 3), the A-discriminant variety has codimension 2 and degree 19.The Nashmatrix has format 11 × 20.Source: §3.7 · pp. 27–28
  • In the (2, 2, 3) example, the Nash equilibrium scheme consists of a point and a line not containing that point.The point is a totally mixed Nash equilibrium; the line also contains totally mixed equilibria.Source: §3.7 · p. 28
  • The example's point on the A-discriminant is not on the one-codimensional part of the Nash discriminant variety, because the 2 × 2 × 2 hyperdeterminant does not vanish there.The last two equations' discriminant is the classical 2 × 2 × 2 hyperdeterminant; its zero locus is the unique hypersurface component of Δ(2, 2, 3) [APS26, Example 3.27].Source: §3.7 · p. 28

4. Mixed Segre embeddings are polynodal

The section studies generic curves cut from the mixed Segre embedding and proves that their projections to each factor are nonsingular for strictly balanced formats. It then relates this geometry to the degree of the Nashmatrix A-discriminant.

4. Mixed Segre embeddings are polynodal

  • The Nashmatrix equations are linear in each player's variables except that of the player whose equation is considered, and the A-discriminant is interpreted as a multigraded Hurwitz locus.The section uses this setup to revisit the degree formula for ∇A.Source: §4 · p. 29
  • For strictly balanced formats, every projection of the generic curve C_i to a factor of P is nonsingular.The result implies that X is α-polynodal.Source: §4 · pp. 30–31
  • For strictly balanced formats, the degree of ∇A equals u1 + ··· + un.Each u_i is specified as a coefficient in the cited formula.Source: §4 · p. 31
  • When d1 = ··· = dn = 2, the section gives the genus of C′_i as 2g(C′_i) − 2 = (n − 3) · ! (n − 1) + (n − 4) · !n.Here !m denotes the number of derangements of m elements.Source: §4 · pp. 31–32

5. The irreducible components of the Nash discriminant variety

The section compares the Nash A-discriminant with ∇A and applies results on BK-tuples to describe the Nash discriminant variety. It establishes the component structure for strictly balanced and on-boundary formats and gives component degree descriptions.

5. The irreducible components of the Nash discriminant variety

  • For balanced formats with at least three players, the section describes the irreducible components of Δ(d) and their degrees, resolving [APS26, Problem 3.8].Its main tools are [Pok26, Corollary 3.6 and Theorem 1.5].Source: §5 · p. 32

5.1. Comparing DA with ∇A.

  • The Nash A-discriminant D_A equals the affine cone over Δ(E), and its irreducible components have the same dimensions and degrees as those of Δ(d).The text identifies D_A with the cone over Δ(E) and states that Δ(d) is a projective cone over Δ(E).Source: §5.1 · pp. 33–34

5.2. Applying Corollary 5.1 to the strictly balanced case.

  • For strictly balanced formats, Δ(d) is an irreducible hypersurface whose degree is given by the coefficient formula in Corollary 5.10.The section establishes that ∇A coincides with Δ(d) in this case.Source: §5.2 · p. 34

5.3. Applying Theorem 5.2 to the on-boundary case.

  • For on-boundary formats, Δ(d) has two irreducible components, of codimensions one and two, with degrees given by the stated formulas.The codimension-two component is identified with ∇A.Source: §5.3 · p. 35
Loading 2610.29889v1…