Pith. sign in

REVIEW 3 major objections 5 minor 53 references

Generalized potential games

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper claims that games derived from detailed-balanced chemical reaction networks are generalized potential games, whose equilibria are minimizers of relative entropy.

desk verdict A real idea—detailed-balanced reaction networks as generalized potential games—with a correct central construction, but the advertised characterization theorems are not ready: Theorem 2.6 is false as stated and Theorem 3.4's converse is unproved. read the letter →

arxiv 1908.06266 v1 pith:7JYRZYI5 submitted 2019-08-17 cs.GT math.APmath.DSmath.OC

M. H. Duong , T. H. Dang-Ha , Q. B. Tang , H. M. Tran This is my paper · ORCID
classification cs.GTmath.APmath.DSmath.OC MSC 91A1091A80
keywords generalizedpotentialgamessimultaneousgradientdissipationsymmetrizablematricesdetailedbalancechemicalreactionnetworksrelativeentropyexponentialconvergence
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

The paper introduces generalized potential games: games whose simultaneous gradient is a nonlinear function of the gradient of a single potential, instead of a weighted linear copy. It shows that such games are characterized by the symmetry of the matrix $D^2\Psi(\xi(w))H(w)$ for some dissipation potential $\Psi$, and that ordinary potential games are the special case where $\Psi$ is quadratic. The main application is a construction from reversible chemical reaction networks: whenever the network has a positive detailed-balance equilibrium, the resulting game is generalized potential, with relative entropy as the potential. A sympathetic reader should care because this transfers the equilibrium machinery of potential games—single-potential optimization and convergence to Nash stationary points—to a much broader class, and makes chemical equilibria appear as Nash equilibria of an explicit game.

What carries the argument

The carrying mechanism is the pair $(\Psi,\Psi^*)$ of Legendre-dual dissipation potentials, whose defining property is that the simultaneous gradient factors as $\xi(w)=D_\zeta\Psi^*(w,\nabla_w E(w))$; this replaces the constant weight matrix of a classical potential game with a state-dependent operator. For the chemical-network example, $\Psi^*$ is a weighted sum over reactions of the log-mean function $\mathfrak l(a,b)$, and $E$ is the relative entropy anchored at the detailed-balance equilibrium; the log-mean function is the exact object that converts the discrete-difference ratio $(w^\alpha-w^\beta)/\log(w^\alpha/w^\beta)$ into the mass-action rate. The symmetrizability criterion that $D^2\Psi(\xi(w))H(w)$ be symmetric links the new definition to the classical Hessian condition and is checkable through cycle conditions of the same type that characterize symmetrizable matrices.

What would settle it

A direct check: for a detailed-balanced network, evaluate both sides of $\xi(w)=H(w)\nabla E(w)$ at several interior states; any mismatch refutes Theorem 3.8. For the characterization, find a game and dissipation potential for which $D^2\Psi(\xi(w))H(w)$ is symmetric but the path integral in (14) depends on the path; that would show the claimed sufficiency direction of Theorem 3.4 does not hold.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is Theorem 3.8: for a reversible reaction network with a positive detailed-balance equilibrium $w_\infty$, the $n$-player game with loss functions (21a)–(21b) is a generalized potential game in the sense of $\xi(w)=D_\zeta\Psi^*(w,\nabla_w E(w))$. The proof exhibits the potential explicitly as the relative entropy $E(w)=\sum_i w_i(\log(w_i/w_i^\infty)-1)$ and a conjugate dissipation $\Psi^*(w,\mu)=\frac12\sum_r \kappa_r\,\mathfrak l(w^{\alpha^r}/w_\infty^{\alpha^r},w^{\beta^r}/w_\infty^{\beta^r})\langle\mu,\alpha^r-\beta^r\rangle^2$, where $\mathfrak l(a,b)=(a-b)/(\log a-\log b)$ is the log-mean function; the detailed-balance identity is exactly what turns $\nabla_\mu\Psi^*(\nabla E)$ into the mass-action rate $\sum_r(k_f w^{\alpha^r}-k_b w^{\beta^r})(\alpha^r-\beta^r)$, which is the simultaneous gradient. The paper further claims that a differentiable game is generalized potential if and only if $D^2\Psi(\xi(w))H(w)$ is symmetric (Theorem 3.4), recovering weighted potential games when $\Psi$ is quadratic, and proves an explicit exponential convergence rate for the special case of a single reversible reaction with any number of species via an explicit entropy-dissipation inequality.

Load-bearing premise

The construction collapses if the reversible reaction network has no positive detailed-balance equilibrium $w_\infty$: equality (26) uses $k_f w_\infty^{\alpha^r}=k_b w_\infty^{\beta^r}$ to identify the dissipation derivative with the simultaneous gradient, so without (20) the game need not be generalized potential.

Editorial extensions

If this is right

  • Every reversible chemical reaction network satisfying detailed balance gives an $n$-player game whose Nash stationary points coincide with critical points of the relative-entropy potential, so projected gradient descent on the loss functions computes chemical equilibria.
  • Generalized potential games inherit the main tool of potential games: equilibrium finding reduces to minimizing a single potential, and the simultaneous gradient vanishes exactly at stationary points of that potential.
  • The symmetry condition that $D^2\Psi(\xi(w))H(w)$ be symmetric gives an intrinsic criterion for whether a differentiable game is generalized potential; for quadratic dissipation it reduces to the classical weighted-potential condition.
  • For a single reversible reaction with an arbitrary number of species, the paper's explicit entropy-dissipation estimate yields a computable exponential decay rate and constants, so the trend to equilibrium is quantitative rather than merely qualitative.

Reading between the lines

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

  • Extrapolating from the continuous proof, the projected-gradient iterates for a single reversible reaction likely inherit an explicit exponential rate, since the discrete dissipation used in the numerical algorithm parallels the entropy-dissipation identity; the paper leaves this as future work.
  • The construction suggests a broader recipe: any game whose simultaneous gradient factors as a state-dependent positive semidefinite operator times the gradient of a functional will be generalized potential, which links the notion to decomposition of vector fields into gradient and Hamiltonian parts.
  • If the detailed-balance assumption fails but a complex-balanced equilibrium still exists, the constructed pair $(E,\Psi^*)$ no longer yields $\xi$, yet the alternative large-deviation dissipation described in Remark 3.10 may still provide a generalized potential structure; testing this on a complex-balanced but not detailed-balanced network would separate the two hypotheses.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper introduces "generalized potential games," in which the simultaneous gradient of the loss functions is written as a nonlinear function of the gradient of a potential through a dissipation potential and its Legendre-Fenchel dual. It claims a necessary and sufficient characterization of such games via a symmetrizability condition on the Hessian, and it applies the framework to games arising from detailed-balanced chemical reaction networks, proving that such games are generalized potential. The paper also states an explicit exponential convergence result for a single reversible reaction and presents numerical experiments with projected gradient descent. The central construction in Theorem 3.8 is plausible and connects the game-theoretic setting to the generalized gradient-flow literature, but the advertised characterization results contain serious gaps and errors.

Significance. If the main claims were correct, the paper would provide a meaningful bridge between potential game theory and generalized gradient flows, and the chemical-reaction application would give a nontrivial class of games beyond classical potential games. The construction in Theorem 3.8, which uses the relative entropy E(w)=Σ w_i(log(w_i/w_i∞)-1) and the log-mean dissipation structure, is mathematically sound and is a genuine strength. The numerical comparisons are useful illustrations. However, the claimed characterizations—both for classical potential games (Theorem 2.6) and for generalized potential games (Theorem 3.4)—are not established as stated, and the Appendix A convergence-rate proof contains a reversed inequality. These issues affect the paper's advertised contributions, not merely its presentation.

major comments (3)
  1. [Section 2, Theorem 2.6] Theorem 2.6 is false as stated. Symmetrizability of H(w) for each w does not imply the existence of a single constant symmetrizer M that makes MH(w) symmetric for all w, which is what condition (3) requires. A concrete counterexample is the two-player game with scalar strategies x,y and losses l1(x,y)=xy and l2(x,y)=(1/2)x^2 y. The Hessian is H(x,y)=[[0,1],[x,0]], which is symmetrizable for each x>0 via D=diag(x,1), but no positive constant diagonal M=diag(α1,α2) can satisfy α1=α2 x for all x. Hence the game is not a potential game despite H being pointwise symmetrizable. The theorem should be replaced by the statement that the game is potential iff there exists a constant diagonal M such that MH(w) is symmetric for all w; the pointwise criterion of Lemma 2.5 is insufficient. This invalidates the claimed contribution (2) as written.
  2. [Section 3.1, Theorem 3.4] The proof of Theorem 3.4 establishes only the necessity of the symmetry condition D²Ψ(ξ(w))H(w) being symmetric. The sufficiency direction, which is essential for the claimed "iff," is not proved. One must show that the 1-form ω = Σ_i D_{s_i}Ψ(ξ(w)) dw_i is closed and hence exact on the strategy domain, so that a potential E satisfying (12) exists. The symmetry condition does give the closure of this form, because ∂_{w_j}(D_{s_i}Ψ(ξ(w))) = [D²Ψ(ξ(w))H(w)]_{ij}, but this step is omitted and the domain assumptions (e.g., simple connectedness) are not stated. Without this argument, Theorem 3.4 is only a necessary condition, not a characterization.
  3. [Appendix A, Step 2, case (ii), Eq. (43)] The lower-bound inequality in case (ii) is reversed. From a_i ≥ L_i one obtains 1/a_i ≤ 1/L_i, and therefore Σ_i α_i²/a_i ≤ Σ_i α_i²/L_i, not ≥. Consequently the claimed bound Λ(a,b) ≥ ∏_i L_i^{α_i} Σ_i α_i²/L_i does not follow. This invalidates the proof of the explicit rate λ in Proposition 3.13 as written. The bound may be repairable by combining the lower bounds a_i ≥ L_i with the upper bounds a_i ≤ α_i min_j M_{ij} coming from the conservation laws, but that repair is not present and the stated inequality is wrong.
minor comments (5)
  1. [Section 3.1, Definition 3.3] The domain of the dissipation potential is written inconsistently: Definition 3.1 uses Ψ:Z×TZ→R, while Definition 3.3 writes Ψ:TZ→R. This should be made uniform.
  2. [Proof of Theorem 3.8] The line "Hence ∇wΨ*(w,µ)=H(w)µ" uses the subscript w incorrectly; the gradient is with respect to the dual variable µ, so it should be ∇_µΨ*(w,µ)=H(w)µ. The same notational confusion appears in the subsequent display.
  3. [Equation (21)] In the displayed loss functions (21a)–(21b), the exponents mix α and β in the second term; for example, (21a) contains w_2^{α_2^r} inside a term involving β, which appears to be a typo and should be checked against the intended definition ξ(w)=Σ_r (k_fw w^{α^r}-k_bw w^{β^r})(α^r-β^r).
  4. [Proposition 3.13] The statement assumes α_1,...,α_m,β_1,...,β_n ∈ [1,∞), but for stoichiometric coefficients nonnegative integers would be more standard; if the continuous range is intended, it should be explicitly justified.
  5. [Throughout] There are several typographical errors and inconsistencies, e.g., "differntiable" in the proof of Theorem 3.4, "generalised" versus "generalized" spellings, and the unnumbered equation after (28) missing a closing parenthesis. A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity; the central generalized-potential construction is a direct, self-contained derivation.

full rationale

The central derivation in Theorem 3.8 is self-contained. The loss functions (21a)-(21b) are defined so that the simultaneous gradient ξ(w) is exactly the reaction rate vector (equation (27)). The paper then constructs E(w) and Ψ∗(w,μ) explicitly in (22)-(23), and the computation (26) uses the detailed-balance identity κ_r = k_fw w∞^{α^r} = k_bw w∞^{β^r} to rewrite the constructed dissipation gradient as the reaction rate. This is an explicit existence proof, not a fitted parameter relabeled as a prediction, and it does not assume the conclusion. The construction is inspired by, but not logically reduced to, the cited gradient-flow literature [Mie11, MPR14]; the decisive algebraic step is performed in the paper itself. Proposition 3.11 cites [DFT17], which has an overlapping author, but that result is background motivation and is not used to prove Theorem 3.8 or the explicit rate in Appendix A; the appendix gives an independent Bakry-Emery proof. The paper does contain real mathematical gaps: Theorem 2.6 is false as stated because pointwise symmetrizability does not imply a single constant diagonal symmetrizer, and the sufficiency direction of Theorem 3.4 is not proved. These are correctness concerns, not circularity: no input is renamed as an output, and no self-citation is load-bearing for the main construction. Hence the circularity score is low, reflecting only the minor self-citation in the background convergence statement.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The derivation rests on the existence of a positive detailed balance equilibrium (imported from Feinberg), on standard convex duality, and on the unproved sufficiency of the symmetrizability condition inside Theorem 3.4. No new entities are introduced; the only hand-chosen numbers are the constants κ_r, which are forced to equal k_fw w∞^{α^r} under detailed balance.

free parameters (1)
  • κ_r = k_fw w∞^{α^r} = k_bw w∞^{β^r}
    The dissipation potential (23) contains constants κ_r; the final equality (26) requires this specific choice, which is determined by the assumed detailed balance equilibrium. It is not fitted to data, but it is part of the construction.
assumptions (4)
  • domain assumption The reaction network has a positive detailed balance equilibrium, unique in its compatibility class (Proposition 3.6).
    Cited from Feinberg's monograph [Fei19]; used in Theorem 3.8 to construct E and in Lemma A.1 for the single-reaction equilibrium.
  • standard math The Legendre-Fenchel duality properties in Lemma 3.2 hold for the dissipation potentials used.
    Standard convex analysis (Fenchel 1949); used to pass between (11) and (12) and to justify the definition.
  • ad hoc to paper The one-form D_sΨ(ξ(w))·dw is closed and hence exact on the domain, which is the sufficiency direction of Theorem 3.4.
    The theorem states an iff but the proof only shows necessity; the path-independence and existence of E are not proved, so the sufficiency is effectively an extra assumption.
  • domain assumption During the evolution (S), the concentrations stay in a compact subset of the positive orthant so that the constant λ in (44) is positive.
    Used in Appendix A Step 2 to bound Λ from below; the published inequality in case (ii) is reversed, so the claimed bound relies on this boundedness in an unspecified way.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Generalized potential games." pith.science (2026). https://pith.science/paper/7JYRZYI5

@misc{pith2026190806266,
  author       = {Pith},
  title        = {Pith review of: Generalized potential games},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7JYRZYI5}},
  note         = {Machine review of arXiv:1908.06266}
}
read the original abstract

In this paper, we introduce a notion of generalized potential games that is inspired by a newly developed theory on generalized gradient flows. More precisely, a game is called generalized potential if the simultaneous gradient of the loss functions is a nonlinear function of the gradient of a potential function. Applications include a class of games arising from chemical reaction networks with detailed balance condition. For this class of games, we prove an explicit exponential convergence to equilibrium for evolution of a single reversible reaction. Moreover, numerical investigations are performed to calculate the equilibrium state of some reversible chemical reactions which give rise to generalized potential games.

Figures

Figures reproduced from arXiv: 1908.06266 by the authors.

Figure 1
Figure 1. Comparing convergence of projected gradient method and numer￾ical solution for different chemical reactions satisfying mass conservation law System (32) has three linearly independent conservation laws, for all t ≥ 0, w1(t) + w4(t) + 2w5(t) + w7(t) + w8(t) = M1 w2(t) + w4(t) + w6(t) + 2w7(t) = M2, w3(t) + w8(t) = M3. (33) By direct computations, for positive initial masses M1, M2, M3 > 0, there exists a unique posit… view at source ↗
Figure 2
Figure 2. Convergence of (a) gradient descend method and (b) numerical solution for chemical reaction that does not satisfy mass conservation law For simplicity, we assume that all reaction rate constants are one. Denote by w1, w2, w3 the concentrations of S1, S2, S3 respectively. We obtain, thanks to the law of mass action w˙ 1 = w1w2 − 3w 3 1 − w 2 1w3 + 3w 2 2 , w˙ 2 = w 3 1 + 2w 2 1w3 − 3w 2 2 , w˙ 3 = w 3 1 − 2w 2 1w3 + … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

53 extracted references · 52 canonical work pages

  1. [1]

    u rich. Birkh \

    L. Ambrosio, N. Gigli, and G. Savare. Gradient Flows: In Metric Spaces and in the Space of Probability Measures . Lectures in Mathematics. ETH Z \"u rich. Birkh \"a user Basel, 2008

    work page 2008

  2. [2]

    A proof of the global attractor conjecture in the single linkage class case

    David F Anderson. A proof of the global attractor conjecture in the single linkage class case. SIAM Journal on Applied Mathematics , 71(4):1487--1508, 2011

    work page 2011

  3. [3]

    Armstrong and J

    M. Armstrong and J. Vickers. Competitive price discrimination. The RAND Journal of Economics , 32(4):579--605, 2001

    work page 2001

  4. [4]

    Baksalary and R

    J.K. Baksalary and R. Kala. Symmetrizers of matrices. Linear Algebra and its Applications , 35:51 -- 62, 1981

    work page 1981

  5. [5]

    Bailey and Georgios Piliouras

    James P. Bailey and Georgios Piliouras. Multi-agent learning in network zero-sum games is a hamiltonian system. In Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems , AAMAS '19, pages 233--241, Richland, SC, 2019. International Foundation for Autonomous Agents and Multiagent Systems

    work page 2019

  6. [6]

    Balduzzi, S

    D. Balduzzi, S. Racaniere, J. Martens, J. Foerster, K. Tuyls, and T. Graepel. The mechanics of n-player differentiable games. In Jennifer Dy and Andreas Krause, editors, Proceedings of the 35th International Conference on Machine Learning , volume 80 of Proceedings of Machine Learning Research , pages 354--363, Stockholmsmässan, Stockholm Sweden, 10--15 J...

    work page 2018

  7. [7]

    Toric dynamical systems

    Gheorghe Craciun, Alicia Dickenstein, Anne Shiu, and Bernd Sturmfels. Toric dynamical systems. Journal of Symbolic Computation , 44(11):1551--1565, 2009

    work page 2009

  8. [8]

    P. H. Calamai and J.J. Mor \'e . Projected gradient methods for linearly constrained problems. Mathematical Programming , 39(1):93--116, Sep 1987

    work page 1987

Show all 53 references
  1. [9]

    E.S.S. Dias, D. Castonguay, and M.C. Dourado. Algorithms and properties for positive symmetrizable matrices. TEMA (S\ a o Carlos) , 17:187 -- 198, 2016

    2016

  2. [10]

    Dittrich and P

    P. Dittrich and P. S. di Fenizio. Chemical organisation theory. Bulletin of Mathematical Biology , 69(4):1199--1231, May 2007

    2007

  3. [11]

    Desvillettes, K

    L. Desvillettes, K. Fellner, and B. Q. Tang. Trend to equilibrium for reaction-diffusion systems arising from complex balanced chemical reaction networks. SIAM Journal on Mathematical Analysis , 49(4):2666--2709, 2017

    2017

  4. [12]

    M.H. Duong. Large deviation and variational approaches to generalized gradient flows . PhD thesis, Department of Mathematics and Computer Science, 9 2014

    2014

  5. [13]

    Feinberg

    M. Feinberg. Foundations of Chemical Reaction Network Theory . Applied Mathematical Sciences. Springer International Publishing, 2019

    2019

  6. [14]

    W. Fenchel. On conjugate convex functions. Canadian Journal of Mathematics , 1(1):73–77, 1949

    1949

  7. [15]

    Fooshee, A

    D. Fooshee, A. Mood, E. Gutman, M. Tavakoli, G. Urban, F. Liu, N. Huynh, D. Van Vranken, and P. Baldi. Deep learning for chemical reaction prediction. Mol. Syst. Des. Eng. , 3:442--452, 2018

    2018

  8. [16]

    Projection methods for linear equality constrained problems, 2004

    Robert M Freund. Projection methods for linear equality constrained problems, 2004. URL: http://mit.ucu.ac.ug/NR/rdonlyres/Sloan-School-of-Management/15-084JSpring2004/ 55C5E7E4-A717-4CFC-8A13-928DBA2F5A9C/0/lec9_project_mthd.pdf. Last visited on 2019/08/14

    2004

  9. [17]

    Fellner and B

    K. Fellner and B. Q. Tang. Convergence to equilibrium of renormalised solutions to nonlinear chemical reaction--diffusion systems. Zeitschrift f \"u r angewandte Mathematik und Physik , 69(3):54, 2018

    2018

  10. [18]

    Goodfellow, J

    I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio. Generative adversarial nets. In Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger, editors, Advances in Neural Information Processing Systems 27 ,...

    2014

  11. [19]

    Heliou, J

    A. Heliou, J. Cohen, and P. Mertikopoulos. Learning with bandit feedback in potential games. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems 30 , pages 6369--6378. Curran ...

    2017

  12. [20]

    J. Z. Hearon. The kinetics of linear systems with special reference to periodic reactions. The bulletin of mathematical biophysics , 15(2):121--141, Jun 1953

    1953

  13. [21]

    J\" u ngel and S

    A. J\" u ngel and S. Schuchnigg. A discrete bakry-emery method and its application to the porous-medium equation. Discrete & Continuous Dynamical Systems - A , 37, 2017

    2017

  14. [22]

    M. A. Kayala and P. F. Baldi. A machine learning approach to predict chemical reactions. In J. Shawe-Taylor, R. S. Zemel, P. L. Bartlett, F. Pereira, and K. Q. Weinberger, editors, Advances in Neural Information Processing Systems 24 , pages 747--755. Curran Associates, Inc., 2011

    2011

  15. [23]

    Letcher, D

    A. Letcher, D. Balduzzi, S. Racani\` e re, J. Martens, J. Foerster, K. Tuyls, and Th. Graepel. Differentiable game mechanics. Journal of Machine Learning Research , 20(84):1--40, 2019

    2019

  16. [24]

    Q. D. La, Y. H. Chew, and B.-. Soong. Potential Game Theory: Applications in Radio Resource Allocation . Springer Publishing Company, Incorporated, 1st edition, 2016

    2016

  17. [25]

    Liero, A

    M. Liero, A. Mielke, M. A. Peletier, and D. R. M. Renger. On microscopic origins of generalized gradient structures. Discrete & Continuous Dynamical Systems - S , 10:1--35, 2017

    2017

  18. [26]

    J. D. Lee, M. Simchowitz, M. I. Jordan, and B. Recht. Gradient descent only converges to minimizers. In Vitaly Feldman, Alexander Rakhlin, and Ohad Shamir, editors, 29th Annual Conference on Learning Theory , volume 49 of Proceedings of Machine Learning Research , pages 1246--...

    2016

  19. [27]

    A. Mielke. A gradient structure for reaction diffusion systems and for energy-drift-diffusion systems. Nonlinearity , 24(4):1329--1346, mar 2011

    2011

  20. [28]

    A. Mielke. On Evolutionary -Convergence for Gradient Systems , pages 187--249. Springer International Publishing, Cham, 2016

    2016

  21. [29]

    Minarchenko

    I. Minarchenko. Search of nash equilibrium in quadratic nonconvex game with weighted potential. volume 2098, pages 291--303, 2018

    2018

  22. [30]

    P. J. Morrison. A paradigm for joined hamiltonian and dissipative systems. Physica D: Nonlinear Phenomena , 18(1):410 -- 419, 1986

    1986

  23. [31]

    Mielke, M

    A. Mielke, M. A. Peletier, and D. R. M. Renger. On the relation between gradient flows and the large-deviation principle, with applications to markov chains and diffusion. Potential Analysis , 41(4):1293--1327, 2014

    2014

  24. [32]

    Mielke, M

    A. Mielke, M. A. Peletier, and M. Renger. A generalization of onsager's reciprocity relations to gradient flows with nonlinear mobility. Journal of Non-Equilibrium Thermodynamics , 41:141--149, 2016

    2016

  25. [33]

    Monderer and L

    D. Monderer and L. S. Shapley. Potential games. Games and economic behavior , 14(1):124--143, 1996

    1996

  26. [34]

    Montanari and A

    A. Montanari and A. Saberi . Convergence to equilibrium in local interaction games. In 2009 50th Annual IEEE Symposium on Foundations of Computer Science , pages 303--312, Oct 2009

    2009

  27. [35]

    The spread of innovations in social networks

    Andrea Montanari and Amin Saberi. The spread of innovations in social networks. Proceedings of the National Academy of Sciences , 107(47):20196--20201, 2010

    2010

  28. [36]

    D. P. Myatt and C. Wallace. Evolution, Teamwork and Collective Action: Production Targets in the Private Provision of Public Goods . The Economic Journal , 119(534):61--90, 12 2008

    2008

  29. [37]

    D. A. Miller and S. W. Zucker. Copositive-plus lemke algorithm solves polymatrix games. Operations Research Letters , 10(5):285 -- 290, 1991

    1991

  30. [38]

    Nisan, T

    N. Nisan, T. Roughgarden, E. Tardos, and V. V. Vazirani. Algorithmic Game Theory . Cambridge University Press, New York, NY, USA, 2007

    2007

  31. [39]

    L. Onsager. Reciprocal relations in irreversible processes. i. Phys. Rev. , 37:405--426, Feb 1931

    1931

  32. [40]

    L. Onsager. Reciprocal relations in irreversible processes. ii. Phys. Rev. , 38:2265--2279, Dec 1931

    1931

  33. [41]

    Ostrovsky and M

    M. Ostrovsky and M. Schwarz. Adoption of standards under uncertainty. The RAND Journal of Economics , 36(4):816--832, 2005

    2005

  34. [42]

    \"O ttinger

    H.C. \"O ttinger. Beyond Equilibrium Thermodynamics . Wiley, 2005

    2005

  35. [43]

    and Winter L

    Dittrich P. and Winter L. Chemical organizations in a toy model of the political system. Advances in Complex Systems , 11(04):609--627, 2008

    2008

  36. [44]

    R. W. Rosenthal. A class of games possessing pure-strategy nash equilibria. International Journal of Game Theory , 2(1):65--67, Dec 1973

    1973

  37. [45]

    G. N. Simm and M. Reiher. Error-controlled exploration of chemical reaction networks with gaussian processes. Journal of Chemical Theory and Computation , 14(10):5238--5248, 2018

    2018

  38. [46]

    Snyman and D.N

    J.A. Snyman and D.N. Wilke. Practical Mathematical Optimization: Basic Optimization Theory and Gradient-Based Algorithms . Springer Optimization and Its Applications. Springer International Publishing, 2018

    2018

  39. [47]

    Schuurmans and M

    D. Schuurmans and M. A Zinkevich. Deep learning games. In D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett, editors, Advances in Neural Information Processing Systems 29 , pages 1678--1686. Curran Associates, Inc., 2016

    2016

  40. [48]

    Veloz, P

    T. Veloz, P. Razeto-Barry, P. Dittrich, and A. Fajardo. Reaction networks and evolutionary game theory. Journal of Mathematical Biology , 68(1):181--206, 2014

    2014

  41. [49]

    Computing a matrix symmetrizer exactly using modified multiple modulus residue arithmetic

    V.ChVenkaiah and S.KSen. Computing a matrix symmetrizer exactly using modified multiple modulus residue arithmetic. Journal of Computational and Applied Mathematics , 21(1):27 -- 40, 1988

    1988

  42. [50]

    Velegol, P

    D. Velegol, P. Suhey, J. Connolly, N. Morrissey, and L. Cook. Chemical game theory. Industrial & Engineering Chemistry Research , 57(41):13593--13607, 2018

    2018

  43. [51]

    Yamamoto

    K. Yamamoto . A Comprehensive Survey of Potential Game Approaches to Wireless Networks . IEICE Transactions on Communications , 98:1804--1823, 2015

    2015

  44. [52]

    Yu and M

    S. Yu and M. Berthod. A game strategy approach for image labeling. Computer Vision and Image Understanding , 61(1):32 -- 37, 1995

    1995

  45. [53]

    Y. Zhou, M. Kantarcioglu, and B. Xi. A survey of game theoretic approach for adversarial machine learning. Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery , 9(3):e1259, 2019

    2019

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.