The GameTheory package for Macaulay2

Chenyang Zhao; Elke Neuhaus; Erin Connelly; Giacomo Maletto; Hannah Tillmann-Morris; Irem Portakal; Vincenzo Galgano; Zhuang He

REVIEW 2 major objections 3 minor 1 cited by

The GameTheory package for Macaulay2

T0 review · 2 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A new Macaulay2 package computes Nash, correlated, dependency, and conditional independence equilibria as polynomial ideals and polytopes.

desk verdict A genuinely useful Macaulay2 package, but the paper misstates the conditional independence equations in Proposition 5.1 and needs a revision before it should be cited as documentation. read the letter →

arxiv 2507.16755 v1 pith:K56TUJAN submitted 2025-07-22 math.AG

Erin Connelly , Vincenzo Galgano , Zhuang He , Giacomo Maletto , Elke Neuhaus , Irem Portakal , Hannah Tillmann-Morris , Chenyang Zhao This is my paper · ORCID

classification math.AG MSC 14-0491A1013P10

keywords gametheoryNashequilibriumcorrelateddependencySpohnvarietyconditionalindependenceMacaulay2polyhedralgeometry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper introduces GameTheory, a package for the Macaulay2 computer algebra system, and presents it as a unified algebraic setting for four equilibrium concepts in finite normal-form games. The package computes totally mixed Nash equilibria from their defining multilinear equations, correlated equilibria as a convex polytope, dependency equilibria as the Spohn variety, and conditional independence equilibria as a saturated intersection of the Spohn variety with a conditional independence model. If the package is correct, researchers can investigate questions that were previously hard to access: the dimension and degree of equilibrium sets, primary decompositions, and fast counts of equilibrium bounds via mixed volume. The paper supports the claim with worked examples including the Bach or Stravinsky game and a three-player game with a unique totally mixed Nash equilibrium.

What carries the argument

The machinery is an algebraic translation dictionary. Each equilibrium concept is rewritten as a variety or polytope inside the probability ring: Nash equilibria as the solutions of multilinear payoff comparisons, correlated equilibria as a convex polytope defined by linear incentive constraints, dependency equilibria as the Spohn variety generated by the 2x2 minors of the Spohn matrices, and conditional independence equilibria as the saturation of the sum of the Spohn ideal and the conditional independence ideal away from the hyperplanes {p=0} and {sum p=0}. Groebner basis and polyhedral computations on these objects produce the package's outputs, including dimension, degree, irreducibility, vertices, and facets.

What would settle it

Run spohnCI on a random (2x2x2) game over a finite field with the CI statement that player 1 is independent of player 3 given player 2; then compute the same variety by directly saturating the sum of spohnIdeal and ciIdeal with respect to the product of all coordinate and sum hyperplanes using a different term order. If the two resulting ideals differ, the experimental saturation strategy is incorrect.

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Extended reading notes

Core claim

The paper's central claim is that four equilibrium notions for finite normal-form games can be treated uniformly as algebraic or polyhedral objects, and that the GameTheory package version 1.0 implements this treatment correctly and usefully. Totally mixed Nash equilibria are the interior solutions of the multilinear equations defining nashEquilibriumIdeal; correlated equilibria are the points of the correlated equilibrium polytope; dependency equilibria are cut out by the 2x2 minors of the Spohn matrices, defining the Spohn variety; and conditional independence equilibria are obtained by intersecting the Spohn variety with a discrete conditional independence model and saturating away the boundary. The package also computes invariants such as dimension, degree, vertices, and facets, and it provides fast mixed-volume counts for the maximal number of totally mixed Nash equilibria.

Load-bearing premise

The load-bearing premise is that the experimentally chosen saturation order in the Spohn CI routine always computes the correct saturated ideal and terminates; the paper gives performance evidence but no correctness proof for that step.

Editorial extensions

If this is right

Totally mixed Nash equilibria can be computed as the solutions of an explicit multilinear system, so users get not just numerical solutions but the ideal's dimension, degree, and primary decomposition.
The correlated equilibrium polytope is available as a polyhedron, giving direct access to its vertices, facets, and f-vector.
For generic games, spohnIdeal reports the Spohn variety's codimension and degree, and users can test irreducibility.
spohnCI removes boundary hyperplanes from the intersection of the Spohn variety and a conditional independence model, so graphical-model restrictions can be imposed on equilibrium computations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next step the authors do not take is to scan random games of a fixed format and measure how often the Spohn CI variety is nonempty; the package makes such a scan straightforward.
The fast mixed-volume routines could be reused as a subroutine for counting or bounding Nash equilibria in larger games where direct polytope mixed-volume computations stall.
Because the package links game-theoretic varieties to conditional independence models, known classification results for graphical models could be translated into statements about which game graphs admit nontrivial dependency equilibria.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear [Colleague],

You should know two things about this paper. First, the GameTheory package is a real, working contribution: it gives a single Macaulay2 interface for Nash, correlated, Spohn/dependency, and Spohn-CI equilibria, with examples that actually run and match known results. Second, the paper as written has a clear mathematical error in Proposition 5.1: equation (4) is not the conditional independence condition. The correct condition is the vanishing of the 2x2 minors, which is exactly what the package's own ciIdeal computes (see Example 5.2). So the implementation is right and the text is wrong.

What's genuinely new here is the integration: bringing the four equilibrium notions into one package with compatible data structures, plus the bridge to GraphicalModels.m2 for CI ideals. That is a convenience tool, not new theory, but it is a solid one. The examples are well chosen, and the computations are reproducible from the shipped code. The authors also get credit for showing the perturbed Bach-Stravinsky game and the reducible Spohn variety, which makes the behavior of the package concrete.

The soft spots are real but manageable. The equation (4) error is the most serious; it is in the central documentation claim. Anyone who reads the paper to learn the definition of CI equilibrium would be misled. That needs to be fixed, not just as a typo: the displayed equation should be the minor equation. The introduction's phrase 'offering new theoretical insights' overstates things; the theory is in Portakal-Sturmfels and related papers. The numberTMNE method leans on an unreviewed co-authored preprint (Abo-Portakal-Sodomaco), and the authors don't provide an independent cross-check for that number. That's a minor concern for a software paper, as long as that method is flagged as depending on a preprint.

The saturation-order issue the reader worried about is not load-bearing. The paper's experimental 'saturate before and after' strategy does not invalidate the output, since the final saturation of the sum is what it is; the order is a performance choice, not a correctness risk.

Who is this for? People who actually compute equilibria and want to do it in Macaulay2, or those studying Spohn varieties computationally. It belongs in a serious referee process, but the referee should request a corrected Proposition 5.1 and a toned-down introduction. I would accept it after minor revision.

Recommendation: send it to review, but make sure the error in equation (4) is fixed before publication.

Referee Report

2 major / 3 minor

Summary. The paper describes GameTheory, a Macaulay2 package (version 1.0, available since Macaulay2 1.25.05) that implements algebraic and polyhedral models for four equilibrium notions in finite normal-form games: Nash equilibria (via nashEquilibriumIdeal), correlated equilibria (via correlatedEquilibria), dependency equilibria (via spohnIdeal and spohnMatrices), and conditional independence equilibria (via ciIdeal and spohnCI). The paper explains the underlying mathematics, presents Macaulay2 session excerpts as examples (Bach-Stravinsky, a 2x2x2 game with a double totally mixed Nash equilibrium, a random 2x2x2 game, and a perturbed 2x2 game), and discusses auxiliary tools such as numberTMNE for mixed-volume computations and intersectWithCImodel for computing Spohn CI ideals. The central claim is that the package lets users construct and analyze the defining ideals and polyhedral models of these equilibria.

Significance. If correct, the package provides a unified, exact symbolic-algebraic alternative to numerical algorithms for equilibrium computation, with reproducible Macaulay2 code and concrete examples that match known results, including the reducible Spohn variety for Bach-Stravinsky, the prime elliptic curve after a generic perturbation, and the degree-2 equilibrium ideal of the 2x2x2 double-point example. The mixed-volume computation via Chern classes is a fast and elegant addition. The paper is essentially a software documentation piece, so its value depends on the accuracy of its mathematical statements; two statements need correction before the documentation can be considered reliable.

major comments (2)

[§5.1, Proposition 5.1 and Eq. (4)] The conditional independence condition is stated incorrectly. The displayed equation p_{iA iB iC} + p_{jA jB iC} − p_{iA jB iC} + p_{jA iB iC} = 0 is not the defining condition for X_A ⊥⊥ X_B | X_C. The correct condition is the vanishing of all 2×2 minors p_{iA iB iC} p_{jA jB iC} − p_{iA jB iC} p_{jA iB iC} = 0, for i_A, j_A ∈ R_A, i_B, j_B ∈ R_B, and i_C ∈ R_C. In addition, the quantifier in Proposition 5.1 includes a variable j_C that does not appear in the displayed equation. The package's own Example 5.2 uses the correct minor equations, so this is a documentation error rather than an implementation error, but it is load-bearing because the paper's abstract and introduction promise to explain the defining ideals of the four equilibrium notions, and Section 5's CI model is introduced through this false equation. Please correct Proposition 5.1, Eq. (4), and the quantifier.
[§4, Theorem 4.3] The statement "If n = d0 = d1 = 2, then the Spohn variety V_X is an elliptic curve" is false as written. The Bach-Stravinsky game, which has n = d0 = d1 = 2, is shown in Example 4.4 to have a reducible Spohn variety with three primary components. The theorem presumably assumes X is generic, as in the surrounding discussion of "generic payoff tensors," but the theorem's hypothesis omits this genericity assumption, creating an internal inconsistency between the stated theorem and the paper's own computation. Please add the genericity hypothesis or otherwise align the statement with the cited source [13, Theorem 8].

minor comments (3)

[§5.3] The remark that saturating each ideal separately before intersecting, and then saturating the sum, is justified only by experimental performance can be strengthened by observing that saturate(saturate(I,W) + saturate(J,W), W) = saturate(I+J, W), so the strategy is mathematically safe. Stating this equality would remove any concern about the lack of a correctness proof.
[§5.1] The quantifier "for all i_A, j_A ∈ R_A, i_B, j_B ∈ R_B, i_C, j_C ∈ R_C" in Proposition 5.1 is inconsistent with Equ. (4), which does not depend on j_C; this should be fixed along with the equation itself.
[§1] The sentence "This discrepancy does not affect the indexing of variables" is somewhat cryptic; a brief example linking one-based vertex labels to zero-based probability variables would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper implements documented equilibrium definitions in Macaulay2 and benchmarks them against independent examples; the flagged Proposition 5.1 and saturation-order issues are correctness risks, not circular reductions.

full rationale

The paper is a software/implementation paper rather than a derivation of new predictions. Each equilibrium notion is implemented from its standard defining equations: nashEquilibriumIdeal returns the multilinear best-response equations from [15, Thm. 6.6] together with the simplex equations; correlatedEquilibria builds the Aumann polytope from inequalities (2); spohnIdeal returns the ideal of 2x2 minors of the Spohn matrices from [13]; and ciIdeal delegates to GraphicalModels.m2's conditionalIndependenceIdeal, an external package. No parameter is fitted and no output is relabeled as a prediction. The central demonstrations are externally checkable: Bach-or-Stravinsky has well-known correlated and Spohn equilibria, and Example 2.2 is McLennan's Exercise 7.2 with the known equilibrium checked by decompose. The self-citations to Portakal-Sturmfels and Portakal-Sendra-Arranz supply definitions and theorems (e.g., codimension/degree of Spohn varieties, vector-bundle mixed volume) that have their own proofs and are not fitted to the package's output; matching those theorems is a check, not a circular justification. Two non-circular issues are flagged for completeness. (1) Proposition 5.1 and Eq. (4) state the CI condition as p_{iA iB iC} + p_{jA jB iC} - p_{iA jB iC} + p_{jA iB iC} = 0, but the correct discrete CI condition is the vanishing of the 2x2 minors p_{iA iB iC} p_{jA jB iC} - p_{iA jB iC} p_{jA iB iC}; the package's Example 5.2 implements the correct minors, so this is a documentation error, not a circular step. (2) The saturation strategy in Section 5.3 (saturate each ideal, then saturate the sum) is explicitly experimental and unproven, and the paper itself warns about runtime; again, this is a correctness/performance caveat, not an input-output circularity. Accordingly the paper deserves a circularity score of 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No parameters are fitted to data; game payoff tensors are user-supplied inputs. The package's correctness rests on several external theorems: Sturmfels' multilinear characterization of Nash equilibria, Bernstein's mixed volume bound, the Portakal-Sturmfels theorems on Spohn varieties, and the vector bundle mixed volume formula from a co-authored preprint. The latter is the least independently grounded assumption. The paper also assumes that the chosen Macaulay2 saturation strategy terminates correctly.

assumptions (7)

standard math Nash equilibria are characterized by multilinear equations (Sturmfels, Theorem 6.6)
Invoked in Section 2.1 for nashEquilibriumIdeal and nashEquilibriumRing.
standard math Bernstein's theorem bounds isolated complex solutions by mixed volume of Newton polytopes
Used in Section 2.2 for the mixed volume upper bound and deltaList.
domain assumption The mixed volume of deltaList(d) equals the coefficient of product h_i^{d_i-1} in product (sum_{j!=i} h_j)^{d_i-1}
From Abo-Portakal-Sodomaco [1, Thm 2.7], a preprint by co-authors, used by numberTMNE. No independent derivation is given in this paper.
domain assumption For generic games, the Spohn variety is irreducible of codimension d0+...+dn-1-n and degree d0*...*dn-1
Portakal-Sturmfels [13, Thm 6], used in Section 4 to validate spohnIdeal outputs.
domain assumption For n=d0=d1=2 the Spohn variety is an elliptic curve; otherwise rational with a map onto (P1)^n
Portakal-Sturmfels [13, Thm 8], quoted in Theorem 4.3.
standard math Conditional independence statements are cut out by 2x2 minors as stated in Proposition 5.1
Sullivant [16], used by ciIdeal. The displayed equation (4) is mistyped as a sum, though the implementation output uses the correct product form.
standard math The global Markov property of an undirected graph yields the CI statements for that graph
Used in Section 5.2 and by GraphicalModels.m2 when a graph is passed to ciIdeal.

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Cite this review

Pith. "Pith review of The GameTheory package for Macaulay2." pith.science (2026). https://pith.science/paper/K56TUJAN

@misc{pith2026250716755,
  author       = {Pith},
  title        = {Pith review of: The GameTheory package for Macaulay2},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K56TUJAN}},
  note         = {Machine review of arXiv:2507.16755}
}

read the original abstract

We describe the GameTheory package version 1.0 for computing equilibria in game theory available since version 1.25.05 of Macaulay2. We briefly explain the four equilibrium notions, Nash, correlated, dependency, and conditional independence, and demonstrate their implementation in the package with examples.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On the dimensions of correlated equilibrium polytopes of generic games
math.CO 2026-08 reject novelty 7.0 of 10

For generic finite games, the correlated equilibrium polytope is claimed to be affinely isomorphic to the polytope of the subgame on essential strategies, which is either full-dimensional or a singleton, but the proof...

Reference graph

Works this paper leans on

16 extracted references · 15 canonical work pages · cited by 1 Pith paper

[1]

A vector bundle approach to nash equilibria

Hirotachi Abo, Irem Portakal, and Luca Sodomaco. A vector bundle approach to nash equilibria. preprint arXiv:2504.03456, 2025

arXiv 2025
[2]

Subjectivity and correlation in randomized strategies

Robert J Aumann. Subjectivity and correlation in randomized strategies. Journal of mathematical Economics , 1(1):67–96, 1974. 12 CONNELLY , GALGANO, HE, MALETTO, NEUHAUS, PORTAKAL, TILLMANN-MORRIS, ZHAO

work page 1974
[3]

Correlated equilibrium as an expression of bayesian rationality

Robert J Aumann. Correlated equilibrium as an expression of bayesian rationality. Econometrica: Journal of the Econometric Society, pages 1–18, 1987

work page 1987
[4]

D. N. Bernshtein. The number of roots of a system of equations. Functional Analysis and Its Applications, 9(3):183–185, Jul 1975

work page 1975
[5]

Combinatorics of correlated equilibria

Marie-Charlotte Brandenburg, Benjamin Hollering, and Irem Portakal. Combinatorics of correlated equilibria. Experimental Mathematics, 34(2):212–224, 2025

work page 2025
[6]

GameTheory: computing equilibria in game theory

Erin Connelly, Vincenzo Galgano, Zhuang He, Lars Kastner, Giacomo Maletto, Elke Neuhaus, Irem Portakal, Hannah Tillmann-Morris, and Chenyang Zhao. GameTheory: computing equilibria in game theory. Version 1.0. AMacaulay2 package available at https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages

work page
[7]

Elliptic curves in game theory

Abhiram Kidambi, Elke Neuhaus, and Irem Portakal. Elliptic curves in game theory. arXiv preprint arXiv:2501.14612, 2025

work page Pith review arXiv 2025
[8]

The maximal number of regular totally mixed nash equilibria

Richard D McKelvey and Andrew McLennan. The maximal number of regular totally mixed nash equilibria. Journal of Economic Theory, 72(2):411–425, 1997

work page 1997

Show all 16 references

[9]

Advanced fixed point theory for economics, volume 25

Andrew McLennan. Advanced fixed point theory for economics, volume 25. Springer, 2018

2018
[10]

Vazirani, editors.Algorithmic Game Theory

Noam Nisan, Tim Roughgarden, ´Eva Tardos, and Vijay V . Vazirani, editors.Algorithmic Game Theory. Cambridge University Press, Cambridge, UK, 2007

2007
[11]

Nash conditional independence curve

Irem Portakal and Javier Sendra-Arranz. Nash conditional independence curve. Journal of Symbolic Computation , 122:102255, 2024

2024
[12]

Game theory of undirected graphical models

Irem Portakal and Javier Sendra-Arranz. Game theory of undirected graphical models. Journal of Algebra , 666:574–606, 2025

2025
[13]

Geometry of dependency equilibria.Rendiconti dell’Istituto di Matematica dell’Universit`a di Trieste: an International Journal of Mathematics, 54, 2022

Irem Portakal and Bernd Sturmfels. Geometry of dependency equilibria.Rendiconti dell’Istituto di Matematica dell’Universit`a di Trieste: an International Journal of Mathematics, 54, 2022

2022
[14]

Dependency equilibria

Wolfgang Spohn. Dependency equilibria. Philosophy of Science, 74(5):775–789, 2007

2007
[15]

Solving Systems of Polynomial Equations, volume 97 of CBMS Regional Conference Series in Mathematics

Bernd Sturmfels. Solving Systems of Polynomial Equations, volume 97 of CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, RI, 2002

2002
[16]

Sullivant

S. Sullivant. Algebraic Statistics, volume 194 of Graduate Studies in Mathematics . American Mathematical Society, Provi- dence, RI, 2018. (E. Connelly) UNIVERSITY OF OSNABR ¨UCK , OSNABR ¨UCK , G ERMANY E-mail address: erin.connelly@uni-osnabrueck.de; ORCID: 0009-0006-0119-38...

2018

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

A vector bundle approach to nash equilibria

Hirotachi Abo, Irem Portakal, and Luca Sodomaco. A vector bundle approach to nash equilibria. preprint arXiv:2504.03456, 2025

arXiv 2025

[2] [2]

Subjectivity and correlation in randomized strategies

Robert J Aumann. Subjectivity and correlation in randomized strategies. Journal of mathematical Economics , 1(1):67–96, 1974. 12 CONNELLY , GALGANO, HE, MALETTO, NEUHAUS, PORTAKAL, TILLMANN-MORRIS, ZHAO

work page 1974

[3] [3]

Correlated equilibrium as an expression of bayesian rationality

Robert J Aumann. Correlated equilibrium as an expression of bayesian rationality. Econometrica: Journal of the Econometric Society, pages 1–18, 1987

work page 1987

[4] [4]

D. N. Bernshtein. The number of roots of a system of equations. Functional Analysis and Its Applications, 9(3):183–185, Jul 1975

work page 1975

[5] [5]

Combinatorics of correlated equilibria

Marie-Charlotte Brandenburg, Benjamin Hollering, and Irem Portakal. Combinatorics of correlated equilibria. Experimental Mathematics, 34(2):212–224, 2025

work page 2025

[6] [6]

GameTheory: computing equilibria in game theory

Erin Connelly, Vincenzo Galgano, Zhuang He, Lars Kastner, Giacomo Maletto, Elke Neuhaus, Irem Portakal, Hannah Tillmann-Morris, and Chenyang Zhao. GameTheory: computing equilibria in game theory. Version 1.0. AMacaulay2 package available at https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages

work page

[7] [7]

Elliptic curves in game theory

Abhiram Kidambi, Elke Neuhaus, and Irem Portakal. Elliptic curves in game theory. arXiv preprint arXiv:2501.14612, 2025

work page Pith review arXiv 2025

[8] [8]

The maximal number of regular totally mixed nash equilibria

Richard D McKelvey and Andrew McLennan. The maximal number of regular totally mixed nash equilibria. Journal of Economic Theory, 72(2):411–425, 1997

work page 1997