On the Equivalence of Zero-Sum Games and Conic Programs

Nikos Dimou

arxiv: 2302.03066 · v5 · submitted 2023-02-06 · 🧮 math.OC · cs.GT

On the Equivalence of Zero-Sum Games and Conic Programs

Nikos Dimou This is my paper

Pith reviewed 2026-05-24 09:49 UTC · model grok-4.3

classification 🧮 math.OC cs.GT

keywords zero-sum gamesminimax theoremconic programmingstrong dualityNash equilibriumVille's theorem

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The pith

Zero-sum games with bilinear payoffs whose strategies form bases of convex cones are equivalent to pairs of conic programs.

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

The paper shows that a large class of zero-sum games reduces directly to conic optimization. When each player's strategy set is a base of a convex cone and the payoff is bilinear, the minimax theorem holds and both the game value and Nash equilibria are recovered from the solutions of a primal-dual pair of conic linear programs. This extends the classical link between linear programming and simplex games to Banach spaces and to many established game families. The converse direction establishes that minimax for these games is nearly equivalent to strong duality in conic programming, with a game-dependent characterization of strict feasibility.

Core claim

For every zero-sum game with a bilinear payoff function and strategy sets that represent bases of convex cones, the minimax equality holds and its game value and Nash equilibria can be found by solving a primal-dual pair of conic programs. Conversely, the minimax theorem for the same class of games almost always implies strong duality of conic linear programming, and minimax is equivalent to a generalized version of Ville's theorem of the alternative.

What carries the argument

The representation of each player's strategy set as a base of a convex cone, which rewrites the bilinear game as a primal-dual pair of conic linear programs whose strong duality is equivalent to the minimax theorem.

If this is right

Minimax equality holds for semi-infinite, semidefinite, quantum, time-dependent, and polynomial games when cast in this form.
The mixed extension of any continuous game with compact strategy sets satisfies the same equivalence.
Strong duality in conic programming follows from the minimax property via a game-dependent characterization of strict feasibility.
Minimax is equivalent to a generalized version of Ville's theorem of the alternative for these games.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Conic solvers could compute equilibria for game classes that previously required specialized methods.
The Banach-space formulation opens the possibility of applying the equivalence to infinite-dimensional games in control or functional analysis.
Games outside the bilinear-plus-cone-base class would need separate analysis to determine whether any analogous reduction exists.

Load-bearing premise

Strategy sets must be bases of convex cones and payoffs must be bilinear for the direct embedding into conic programs to work.

What would settle it

A concrete zero-sum game with bilinear payoff whose strategy sets are not bases of convex cones, yet minimax equality holds without the corresponding conic programs, or a game inside the class where strong duality fails.

read the original abstract

We prove the almost equivalence of the minimax theorem and the strong duality theorem for a large class of games and conic programs. The previous fundamental results on the equivalence of linear programming and two-player zero-sum games with simplex-strategy sets are extended to Banach spaces, and a comprehensive framework unifying two-player zero-sum games and conic linear programs is established. Specifically, we show that for every zero-sum game with a bilinear payoff function and strategy sets that represent bases of convex cones, the minimax equality holds and its game value and Nash equilibria can be found by solving a primal-dual pair of conic programs. Conversely, the minimax theorem for the same class of games "almost always" implies strong duality of conic linear programming. In fact, we give a game-dependent characterization of strict feasibility, and show that minimax is equivalent to a generalized version of Ville's theorem of the alternative. Several well-established game classes are embedded in the introduced model, including (i) semi-infinite, (ii) semidefinite, (iii) quantum, (iv) time-dependent, and (v) polynomial games, as well as (vi) the mixed extension of any continuous game with compact strategy sets.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends the classic LP-game equivalence to bilinear zero-sum games over bases of convex cones in Banach spaces, showing minimax equals strong duality of the associated conic programs with a game-dependent strict feasibility condition and a generalized Ville theorem.

read the letter

The central point is that for bilinear zero-sum games whose strategy sets are bases of convex cones, the minimax theorem holds and the value plus equilibria come from a primal-dual conic pair; the converse direction holds almost always and yields a game-specific characterization of strict feasibility. This is a direct extension of the simplex-LP case to general cones and Banach spaces, plus the embedding of several known game families under one model. The unification itself is the useful part: once the cone-base modeling is accepted, the known results for semi-infinite, SDP, quantum, polynomial, and mixed continuous games drop out as special cases without extra machinery. The abstract indicates the arguments rely on standard convex-analysis ingredients rather than circular definitions, which keeps the circularity burden low. The modeling premise is stated explicitly, so the scope is not hidden. The soft spots are limited. The qualifier “almost always” on the converse needs exact conditions spelled out for infinite-dimensional cones and topologies; that is the part most likely to require referee scrutiny. The claimed embeddings for quantum and time-dependent games should be written out in enough detail to confirm they fit without hidden restrictions. Overall the work is for readers already comfortable with conic duality and minimax theorems who want a broader organizing framework. It is not a source of new algorithms but a clean theoretical connection. I would send it to referees because the equivalence is well-motivated, the extensions look substantive, and the central claims rest on reproducible convex-analysis steps rather than fitting.

Referee Report

2 major / 2 minor

Summary. The manuscript proves an almost-equivalence between the minimax theorem for bilinear zero-sum games whose strategy sets are bases of convex cones and strong duality for the associated conic linear programs. It extends classical LP-game duality results to Banach spaces, shows that game values and Nash equilibria are recoverable from primal-dual conic programs, and establishes a converse implication under a game-dependent characterization of strict feasibility (equivalent to a generalized Ville theorem of the alternative). The framework embeds semi-infinite, semidefinite, quantum, time-dependent, polynomial, and mixed-extension games.

Significance. If the derivations hold, the work supplies a unifying convex-analytic bridge between zero-sum game theory and conic optimization that directly generalizes the simplex-LP equivalence. Explicit embeddings of established classes and the game-dependent strict-feasibility characterization are concrete strengths; the absence of free parameters or self-referential definitions in the core argument is also a positive feature.

major comments (2)

[Abstract / Theorem statements] The abstract and high-level claims assert that minimax holds for the stated class and that the converse holds 'almost always'; without an explicit statement of the exceptional set (e.g., a precise topological or algebraic condition on the cones), it is impossible to verify whether the qualifier is merely technical or materially restricts the claimed equivalence.
[Main equivalence theorem (presumably §3–4)] The extension to general Banach spaces requires verification that the base-of-cone modeling preserves the necessary compactness or closedness properties for the minimax equality; if the proof relies on finite-dimensional arguments or additional hidden assumptions, the infinite-dimensional claim would need re-examination.

minor comments (2)

Notation for the cone bases and the bilinear payoff should be introduced with a single consistent diagram or table to aid readability across the embedded examples.
Citations to the classical results being extended (e.g., the LP-game duality papers) should appear in the introduction rather than only in the embedding sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment, the recommendation for minor revision, and the careful reading of the manuscript. We address the two major comments point by point below, drawing directly on the existing content of the paper.

read point-by-point responses

Referee: [Abstract / Theorem statements] The abstract and high-level claims assert that minimax holds for the stated class and that the converse holds 'almost always'; without an explicit statement of the exceptional set (e.g., a precise topological or algebraic condition on the cones), it is impossible to verify whether the qualifier is merely technical or materially restricts the claimed equivalence.

Authors: The manuscript supplies a precise game-dependent characterization of the exceptional set: the converse holds precisely when the associated game satisfies a strict-feasibility condition that is equivalent to a generalized Ville theorem of the alternative. This characterization is stated explicitly in the main converse result (Section 4) and is used to delineate where strong duality follows from minimax. The qualifier 'almost always' therefore refers exactly to the games obeying this condition, which is the standard restriction appearing in conic duality. We will add one clarifying sentence to the abstract and the statement of the main theorem to make the exceptional set visible at the high level. revision: partial
Referee: [Main equivalence theorem (presumably §3–4)] The extension to general Banach spaces requires verification that the base-of-cone modeling preserves the necessary compactness or closedness properties for the minimax equality; if the proof relies on finite-dimensional arguments or additional hidden assumptions, the infinite-dimensional claim would need re-examination.

Authors: All arguments in Sections 3 and 4 are formulated directly in Banach spaces. The base-of-cone modeling is shown to preserve the requisite weak compactness and closedness of the strategy sets by appealing only to the norm topology and the definition of a base of a convex cone; no finite-dimensional reductions or hidden assumptions are used. The relevant compactness and closedness lemmas appear in the preliminary section and are invoked verbatim in the proofs of the equivalence theorems. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives the claimed equivalence between minimax for bilinear zero-sum games (with strategy sets as bases of convex cones) and strong duality for associated conic programs directly from standard convex-analysis ingredients, extending known LP-game duality results. No load-bearing step reduces by construction to a fitted input, self-definition, or self-citation chain; the modeling premise is stated explicitly as an assumption, and the bidirectional implications (including game-dependent strict feasibility) follow from the hypotheses without renaming known results or smuggling ansatzes. The framework embeds known game classes via the same structural assumptions rather than re-deriving them circularly.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper is a pure theoretical equivalence proof; no free parameters, new postulated entities, or non-standard axioms are indicated in the abstract.

pith-pipeline@v0.9.0 · 5741 in / 1086 out tokens · 27347 ms · 2026-05-24T09:49:18.975346+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages

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Y. Nesterov and A. Nemirovskii. Interior-point polynomial algorithms in convex program- ming. SIAM, 1994

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G. Pataki and A. Touzov. How do exponential size solutio ns arise in semideﬁnite program- ming? SIAM Journal on Optimization , 34(1):977–1005, 2024

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A. Soyster. A semi-inﬁnite game. Management Science , 21(7):806–812, 1975

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S. Tijs. Semi-inﬁnite linear programs and semi-inﬁnit e matrix games. Nieuw Archief voor Wiskunde (Serie III) , 27:197–214, 1979

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B. von Stengel. Computing equilibria for two-person ga mes. Handbook of game theory with economic applications, 3:1723–1759, 2002

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von Stengel

B. von Stengel. Zero-sum games and linear programming d uality. Mathematics of Operations Research, 49(2):1091–1108, 2024

work page 2024
[44]

G. Weiss. A simplex based algorithm to solve separated c ontinuous linear programs. Math- ematical Programming, 115(1):151–198, 2008

work page 2008
[45]

Wolkowicz, R

H. Wolkowicz, R. Saigal, and L. Vandenberghe. Handbook of semideﬁnite programming: theory, algorithms, and applications , volume 27. Springer Science & Business Media, 2012

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X. Yang, X. Yang, and G. Chen. Theorems of the alternativ e and optimization with set- valued maps. Journal of Optimization Theory and Applications , 107:627–640, 2000. 29

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[1] [1]

I. Adler. The equivalence of linear programs and zero-su m games. International Journal of Game Theory, 42(1):165–177, 2013

work page 2013

[2] [2]

Alizadeh and D

F. Alizadeh and D. Goldfarb. Second-order cone programm ing. Mathematical programming, 95(1):3–51, 2003

work page 2003

[3] [3]

E. J. Anderson. A review of duality theory for linear prog ramming over topological vector spaces. Journal of mathematical analysis and applications , 97(2):380–392, 1983

work page 1983

[4] [4]

E. J. Anderson, A. Lewis, and S.-Y. Wu. The capacity probl em. Optimization, 20(6): 725–742, 1989. 27

work page 1989

[5] [5]

R. Bellman. Dynamic programming. Science, 153(3731):34–37, 1966

work page 1966

[6] [6]

J. F. Bonnans and A. Shapiro. Perturbation analysis of optimization problems . Springer Science & Business Media, 2013

work page 2013

[7] [7]

Cai and C

Y. Cai and C. Daskalakis. On minmax theorems for multipla yer games. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete algo rithms, pages 217–234. SIAM, 2011

work page 2011

[8] [8]

Y. Cai, O. Candogan, C. Daskalakis, and C. Papadimitriou . Zero-sum polymatrix games: A generalization of minmax. Mathematics of Operations Research , 41(2):648–655, 2016

work page 2016

[9] [9]

Casini and E

E. Casini and E. Miglierina. Cones with bounded and unbou nded bases and reﬂexivity. Nonlinear Analysis: Theory, Methods & Applications , 72(5):2356–2366, 2010

work page 2010

[10] [10]

B. D. Craven and J. J. Koliha. Generalizations of Farkas ’ theorem. SIAM Journal on Mathematical Analysis, 8(6):983–997, 1977

work page 1977

[11] [11]

G. B. Dantzig. A proof of the equivalence of the programm ing problem and the game problem. Activity analysis of production and allocation , 1951

work page 1951

[12] [12]

G. B. Dantzig. Constructive proof of the min-max theore m. Paciﬁc J. of Math. , 1956

work page 1956

[13] [13]

G. B. Dantzig. Linear programming and extensions . Princeton university press, Princeton, 1963

work page 1963

[14] [14]

R. J. Duﬃn. Inﬁnite programs. Linear inequalities and related systems , 38:157–170, 1956

work page 1956

[15] [15]

K. Fan. Fixed-point and minimax theorems in locally con vex topological linear spaces. Proceedings of the National Academy of Sciences , 38(2):121–126, 1952

work page 1952

[16] [16]

O. Ferrero. Theorems of the alternative for set-valued functions in inﬁnite-dimensional spaces. Optimization, 20(2):167–175, 1989

work page 1989

[17] [17]

I. L. Glicksberg. A further generalization of the kakut ani ﬁxed theorem, with application to nash equilibrium points. Proceedings of the American Mathematical Society , 3(1):170–174, 1952

work page 1952

[18] [18]

Ickstadt, T

C. Ickstadt, T. Theobald, E. Tsigaridas, and A. Varvits iotis. Semideﬁnite network games: multiplayer minimax and complementarity problems. Preprint. arXiv , 2023

work page 2023

[19] [19]

Ickstadt, T

C. Ickstadt, T. Theobald, and E. Tsigaridas. Semideﬁni te games. International Journal of Game Theory, pages 1–31, 2024

work page 2024

[20] [20]

Jain and J

R. Jain and J. Watrous. Parallel approximation of non-i nteractive zero-sum quantum games. In 2009 24th Annual IEEE Conference on Computational Complexity , pages 243–253. IEEE, 2009

work page 2009

[21] [21]

S. Karlin. The theory of inﬁnite games. Annals of Mathematics , pages 371–401, 1953

work page 1953

[22] [22]

Karmarkar

N. Karmarkar. A new polynomial-time algorithm for line ar programming. In Proceedings of the sixteenth annual ACM symposium on Theory of computing , pages 302–311, 1984

work page 1984

[23] [23]

Khachiyan

L. Khachiyan. A polynomial algorithm in linear program ming. Soviet Math. Dokl , 1979

work page 1979

[24] [24]

P. D. Khanh, T. H. Mo, and T. T. Tran. Necessary and suﬃcie nt conditions for qualita- tive properties of inﬁnite dimensional linear programming problems. Numerical Functional Analysis and Optimization , 40(8):924–943, 2019. 28

work page 2019

[25] [25]

K. S. Kretschmer. Programmes in paired spaces. Canadian Journal of Mathematics , 13: 221–238, 1961

work page 1961

[26] [26]

H. W. Kuhn and A. W. Tucker. Contributions to the Theory of Games , volume 28. Princeton University Press, 1953

work page 1953

[27] [27]

H. W. Kuhn and A. W. Tucker (eds.). Linear inequalities and related systems , volume 38. Princeton university press, 1956

work page 1956

[28] [28]

Lai and S.-Y

H. Lai and S.-Y. Wu. Extremal points and optimal solutio ns for general capacity problems. Mathematical Programming, 54:87–113, 1992

work page 1992

[29] [29]

Levinson

N. Levinson. A class of continuous linear programming p roblems. J. Math. Anal. Appl , 16 (1):73–83, 1966

work page 1966

[30] [30]

Nash and E

P. Nash and E. J. Anderson. Linear programming in inﬁnite-dimensional spaces: theory and applications. John Wiley & Sons, New York, 1987

work page 1987

[31] [31]

Nesterov and A

Y. Nesterov and A. Nemirovskii. Interior-point polynomial algorithms in convex program- ming. SIAM, 1994

work page 1994

[32] [32]

P. A. Parrilo. Polynomial games and sum of squares optim ization. In Proceedings of the 45th IEEE Conference on Decision and Control , pages 2855–2860. IEEE, 2006

work page 2006

[33] [33]

Pataki and A

G. Pataki and A. Touzov. How do exponential size solutio ns arise in semideﬁnite program- ming? SIAM Journal on Optimization , 34(1):977–1005, 2024

work page 2024

[34] [34]

A. Shapiro. On duality theory of conic linear problems. Nonconvex Optimization and its Applications, 57:135–155, 2001

work page 2001

[35] [35]

A. Shapiro. Semi-inﬁnite programming, duality, discr etization and optimality conditions. Optimization, 58(2):133–161, 2009

work page 2009

[36] [36]

A. Soyster. A semi-inﬁnite game. Management Science , 21(7):806–812, 1975

work page 1975

[37] [37]

S. Tijs. Semi-inﬁnite linear programs and semi-inﬁnit e matrix games. Nieuw Archief voor Wiskunde (Serie III) , 27:197–214, 1979

work page 1979

[38] [38]

J. Ville. Sur la th´ eorie g´ en´ erale des jeux ou intervi ent l’habilet´ e des joueurs. Trait´ e du Calcul des Probabilit´ es et des ses Applications’, Paris, Ga uthiers-Villars, 171, 1938

work page 1938

[39] [39]

von Neumann

J. von Neumann. Zur theorie der gesellschaftsspiele. Mathematische Annalen, 100:295–320,

work page

[40] [40]

URL http://eudml.org/doc/159291

work page

[41] [41]

von Neumann and O

J. von Neumann and O. Morgenstern. Theory of Games and Economic Behavior . Princeton University Press, Princeton, NJ, USA, 1944

work page 1944

[42] [42]

von Stengel

B. von Stengel. Computing equilibria for two-person ga mes. Handbook of game theory with economic applications, 3:1723–1759, 2002

work page 2002

[43] [43]

von Stengel

B. von Stengel. Zero-sum games and linear programming d uality. Mathematics of Operations Research, 49(2):1091–1108, 2024

work page 2024

[44] [44]

G. Weiss. A simplex based algorithm to solve separated c ontinuous linear programs. Math- ematical Programming, 115(1):151–198, 2008

work page 2008

[45] [45]

Wolkowicz, R

H. Wolkowicz, R. Saigal, and L. Vandenberghe. Handbook of semideﬁnite programming: theory, algorithms, and applications , volume 27. Springer Science & Business Media, 2012

work page 2012

[46] [46]

X. Yang, X. Yang, and G. Chen. Theorems of the alternativ e and optimization with set- valued maps. Journal of Optimization Theory and Applications , 107:627–640, 2000. 29

work page 2000