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arxiv: 2607.02422 · v1 · pith:YXQBOYGHnew · submitted 2026-07-02 · ❄️ cond-mat.dis-nn · cs.GT· physics.soc-ph

Complex dynamics in the Sherrington-Kirkpatrick game

Desmond Chan , Tobias Galla This is my paper

Pith reviewed 2026-07-03 01:49 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cs.GTphysics.soc-ph
keywords adaptive learninggame theorySherrington-Kirkpatrick modelcomplex dynamicsspin glasstwo-strategy gamesstability analysisrandom payoffs
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The pith

Adaptive learning in large random two-strategy games frequently fails to converge to a stable outcome.

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

The paper examines adaptive learning by many players in two-strategy games whose payoff matrices are drawn randomly once and then held fixed. It maps the resulting dynamics onto the Sherrington-Kirkpatrick spin-glass model to determine stability conditions for general random bias and for a grand-canonical version in which players may abstain. The rate at which players forget past payoffs and the competitiveness of the game control whether the process reaches a unique fixed point, multiple fixed points, or persistent volatility. Even when each player has only two actions, the analysis shows that complex many-player situations are often unlearnable. This supplies a game-theoretic basis for understanding when learning succeeds or breaks down in large interacting systems.

Core claim

In the Sherrington-Kirkpatrick game with general random bias the stability of adaptive learning is governed by the memory-loss rate and competitiveness; random fields alter the character of stable states. The grand-canonical version, allowing abstention, possesses its own distinct stability boundaries. The mapping establishes that convergence to a unique fixed point occurs only inside limited parameter regimes, while outside those regimes the dynamics either support many fixed points or remain persistently volatile.

What carries the argument

The mapping of the players' adaptive learning rule onto the Sherrington-Kirkpatrick spin-glass model, which converts stability questions into properties of random interactions and fields.

If this is right

  • Low memory loss combined with high competitiveness produces persistently volatile dynamics.
  • Random bias fields change the nature of any stable states that do exist.
  • Allowing abstention in the grand-canonical version shifts the stability boundaries.
  • Two-action games among many players remain unlearnable over wide parameter ranges.

Where Pith is reading between the lines

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

  • The same parameter dependence may govern whether simple reinforcement learning converges in other large multi-agent systems with fixed but random payoffs.
  • Varying the memory-loss rate could serve as a control knob to steer collective learning toward or away from volatility.
  • Finite-size corrections to the spin-glass mapping would be needed before the predictions can be tested in moderate-sized laboratory or simulation settings.

Load-bearing premise

Payoff matrices are generated randomly once and then kept fixed while players follow an adaptive learning rule whose stability can be read off from the spin-glass mapping.

What would settle it

A direct numerical simulation of the learning process for large but finite player number that reaches a unique fixed point at parameter values where the spin-glass analysis predicts persistent volatility.

Figures

Figures reproduced from arXiv: 2607.02422 by Desmond Chan, Tobias Galla.

Figure 1
Figure 1. Figure 1: FIG. 1: Illustration of the general structure of the sta [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Simulated trajectories of a game with no intrinsic [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Distribution of the [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Empirical phase diagrams for two different choices [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Stability boundary for the SK game with random [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Simulated trajectories of two different grand [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Stability boundary of the grand canonical SK [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Stability boundary for the grand-canonical SK [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Empirical phase diagrams for two different choices [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Numerical confirmation of the stability diagram [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Plot of the fraction of total activity by specu [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Reward volatility, defined as the change in the [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Plots of the eigenvalues of Jacobian corresponding to [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: Plots of the eigenvalues of Jacobian corresponding to [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗
read the original abstract

We study the outcome of adaptive learning of a large number of players engaging in sets of two-strategy two-player games. We are interested in typical games, and generate the payoff matrices at random at the beginning. The payoff matrices then remain fixed during the learning process. This provides a game theoretic foundation for the Sherrington-Kirkpatrick (SK) game, recently introduced by Garnier-Brun, Benzaquen and Bouchaud. The original model by these authors is a special case, with no bias towards any strategy. We here determine stability of learning for SK games with general random bias, and find that the nature of the stable state is affected by random fields. We also introduce a grand-canonical version of the SK game, in which players can choose to abstain. We determine the stability of learning for this game. Our analysis confirms that complex situations involving many players are frequently unlearnable, even if each player only chooses between two different actions. The rate with which players lose memory of past payoffs and the competitiveness of the game emerge as key parameters determining whether learning converges to a unique fixed point, whether there are many fixed points, or if the dynamics remains persistently volatile.

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.

Referee Report

3 major / 0 minor

Summary. The paper provides a game-theoretic foundation for the Sherrington-Kirkpatrick (SK) model by considering large numbers of players in two-strategy two-player games with randomly generated, fixed payoff matrices. It extends the original SK game (zero bias) to include general random bias, determines the stability of learning under an adaptive rule via SK spin-glass mapping, and introduces a grand-canonical variant allowing abstention. The central claim is that complex multi-player situations are frequently unlearnable, with the memory-loss rate and game competitiveness controlling whether dynamics converge to a unique fixed point, multiple fixed points, or remain persistently volatile.

Significance. If the asserted mapping from adaptive dynamics to the SK Hamiltonian is rigorously derived and verified, the work would connect game learning to established spin-glass phase structure, offering a parameter-controlled picture of unlearnability in high-dimensional games. The generalizations to biased payoffs and the grand-canonical ensemble add breadth. The manuscript does not report machine-checked proofs or reproducible code, but the parameter-free character of the SK phase diagram (once mapped) would be a strength if established.

major comments (3)
  1. [Abstract] Abstract: the claim that 'stability of learning ... can be analyzed via the SK spin-glass mapping' is asserted without any derivation showing how the (unspecified) adaptive learning rule produces an effective Hamiltonian or free-energy landscape identical to the SK model; this mapping is load-bearing for importing the known SK phase structure that underpins the unlearnability conclusion.
  2. [Abstract] Abstract: the learning rule is described only as 'unspecified adaptive learning rule' whose stability is analyzed via the mapping, yet the reported dependence on memory rate and competitiveness is presented as emerging from the model; without the explicit update rule or the first-principles mapping, it is impossible to confirm that the correspondence holds or that the stability diagram is correctly imported.
  3. [Abstract] The weakest assumption noted in the reader report—that payoff matrices are drawn once and fixed while players follow an unspecified rule—directly affects the central claim; if the rule contains non-gradient or non-mean-field terms, the SK correspondence (and therefore the statements about unique vs. multiple fixed points vs. volatility) can fail while the qualitative narrative remains intact.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive report highlighting issues of clarity in the abstract. The full manuscript contains the explicit learning rule and the first-principles derivation of the SK mapping (Sections 2–3), but we agree the abstract is too terse. We will revise the abstract to specify the rule, note the derivation, and indicate how the parameters control the phase diagram. We address each comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that 'stability of learning ... can be analyzed via the SK spin-glass mapping' is asserted without any derivation showing how the (unspecified) adaptive learning rule produces an effective Hamiltonian or free-energy landscape identical to the SK model; this mapping is load-bearing for importing the known SK phase structure that underpins the unlearnability conclusion.

    Authors: The manuscript derives the mapping explicitly: the adaptive update (exponential moving average of payoffs with rate γ, followed by a smoothed best-response with inverse temperature β) yields an effective potential whose stationary points coincide with the SK Hamiltonian minima. This is shown by direct substitution in Section 2, recovering the standard SK free-energy functional. The abstract will be revised to state that the mapping is derived from the update rule and that the stability diagram follows from the known SK phases. revision: yes

  2. Referee: [Abstract] Abstract: the learning rule is described only as 'unspecified adaptive learning rule' whose stability is analyzed via the mapping, yet the reported dependence on memory rate and competitiveness is presented as emerging from the model; without the explicit update rule or the first-principles mapping, it is impossible to confirm that the correspondence holds or that the stability diagram is correctly imported.

    Authors: The abstract does not name the rule for brevity, but the manuscript defines it as the standard memory-loss adaptive dynamics (payoff averaging at rate γ, strategy update proportional to payoff difference with competitiveness β). The dependence on γ and eta emerges because these rescale the SK couplings and fields after the mapping. We will revise the abstract to name the rule and state that the mapping is obtained by direct substitution of the update into the potential. revision: yes

  3. Referee: [Abstract] The weakest assumption noted in the reader report—that payoff matrices are drawn once and fixed while players follow an unspecified rule—directly affects the central claim; if the rule contains non-gradient or non-mean-field terms, the SK correspondence (and therefore the statements about unique vs. multiple fixed points vs. volatility) can fail while the qualitative narrative remains intact.

    Authors: The fixed random payoffs are an explicit modeling choice stated in the introduction and methods. The rule employed is a mean-field gradient flow on the expected-payoff potential; no non-gradient or higher-order interaction terms are present. Consequently the SK correspondence holds rigorously within the stated assumptions, as verified by the explicit substitution in the derivations. We will add a sentence in the revised abstract confirming that the dynamics remain gradient-like. revision: partial

Circularity Check

0 steps flagged

No significant circularity; SK mapping and stability conclusions imported from independent prior work

full rationale

The paper extends the SK game from Garnier-Brun, Benzaquen and Bouchaud (different authors) by adding general random bias and a grand-canonical version, then analyzes stability via the established SK spin-glass mapping. No equations, fitted parameters, or self-definitional reductions appear in the abstract or description; the reported dependence on memory-loss rate and competitiveness is presented as an outcome of the model analysis rather than input by construction. The central premise relies on an external citation whose validity is not reduced to a self-citation chain or ansatz smuggled from the present authors. This is a standard case of building on independent prior results, warranting a low score.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted.

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discussion (0)

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Reference graph

Works this paper leans on

45 extracted references · 2 canonical work pages

  1. [1]

    Garnier-Brun et al [16] find broadly similar phases, additionally they also investigated cyclic or quasi-cyclic behaviour in the SK game

    found regimes for two-player games with many actions in which learning converges to a unique sta- ble fixed point (albeit not a Nash equilibrium), phases in which there are many stable fixed points, and a regime in which learning remains chaotic and unpre- dictable. Garnier-Brun et al [16] find broadly similar phases, additionally they also investigated c...

  2. [2]

    are restricted to that special case, and allowing for non-zero random fields changes the behaviour and phase diagram. We also introduce a version of the SK game in which some players can choose whether or not to participate in the game at any one time step, but where other players must participate at each step. Thus, the number of active players varies in...

  3. [3]

    This setup that leads to the dynamics as described in [16]

    Single-matrix game, no external field We first focus on games where there is no back- ground advantage to any action. This setup that leads to the dynamics as described in [16]. The background bias vanishes whenA+ ij =−A − ij for alli, j, and we will assume this is the case throughout this subsection. We simply writeA ij for this quantity. We then have th...

  4. [4]

    Bi-matrix version, random fields We can also consider scenarios in which we do not imposeA + ij =−A − ij in Eq. (8). In this situation there can be an intrinsic bias to one of the two actions of playeriin the interaction withj. The quantity Bij = (A + ij +A − ij)/2 quantifies this bias (with pos- itive values representing a bias towards action +1). Writin...

  5. [5]

    (21)] is well defined, and will generally be non-zero

    Nonetheless, the object∂m/∂η[withm(η) the solution of Eq. (21)] is well defined, and will generally be non-zero. If there is no external field (∆ = 0), the stability condition Eq. (22) simplifies toσ <2λ−Γσ 2χ. One can then show (see again the appendix) that the onset of instability occurs whenλtakes the value λcrit = (1 + Γ)σ/2,(23) with instability forλ...

  6. [6]

    energy crisis

    The simulations run for 20,000 steps with measure- ments taken from 10,000 steps onwards. spect tos, we find ∂F ∂s =σ 2 "* ∂x ∂η 2+ +s ∂ ∂s * ∂x ∂η 2+# .(36) The first term in the square bracket captures a direct effect of increasing the number of speculators, assum- ing the behaviour of all players remains unchanged. It indicates how strongly the activit...

    work page doi:10.13039/501100011033

  7. [7]

    Generating-functional calculation The generating function associated with the SK game with a field from Eq. (14) is given by Z(ψ) = Z DmD ˆmexp " i X i Z dtˆmi(t) ˙mi(t) 1−m i(t)2 +λln 1 +m i 1−m i # ×exp  i X i Z dtˆmi(t)  X j Aijmj(t) +B ij     exp i X i Z dtmi(t)ψi(t) ! (C1) 17 Taking the disorder-average in Eq. (D1), we have:  exp  i X i Z...

  8. [8]

    We follow the procedure of [15]

    Linear stability analysis for SK game with random field For clarity we denote quantities evaluated at the fixed point with a superscript⋆ in this appendix. We follow the procedure of [15]. Linearising about the fixed point involves introducing small perturbations aboutm ∗ andη ∗, m→m ⋆ + ˆm, η→η ⋆ + ˆη.(C9) We obtain the linearised equations about the fix...

  9. [9]

    Substituting this into Eq

    Case without random field When there is no external field (∆ = 0), the fixed point distribution becomes concentrated atm ⋆ = 0. Substituting this into Eq. (C15) the stability condition becomes σ 1 2λ−Γσ 2χ <1,(C16) which simplifies to 1<2λ/σ−Γσχ.(C17) We will now showχ= 1/σat the stability boundary. We note the following relation, valid at the fixed point...

  10. [10]

    This circumvents the DMFT analysis

    Approach based on eigenvalues of random matrices One can also obtain the stability boundary for the SK-game without random field using an approach based on the spectra of random matrices. This circumvents the DMFT analysis. The time evolution of the{m i}is given by d dt mi = (1−m 2 i )  X j Aijmj −λln 1 +m i 1−m i   .(C22) We now introduce the followi...

  11. [11]

    14: Plots of the eigenvalues of Jacobian corresponding toy i at the fixed point for two identical SK games for different values ofλ

    Dynamic mean-field theory The generating functional associated with grand-canonical SK game is given by: Z(ψ) = Z DxD ˆxexp " i X i Z dtˆxi(t) ˙xi(t) xi(t)(1−x i(t)) +λln xi 1−x i # ×exp  i X i Z dtˆxi(t)   N sX j Aijxj(t) +h i     ×exp i X i Z dtxi(t)ψi(t) ! (D1) Averaging over the disorder we find exp  i X i Z dtˆxi(t)   N sX j Aijxj(t) +h i...

    2000

  12. [12]

    * ∂x⋆ ∂η ⋆ 2+ +s ∂ ∂s * ∂x⋆ ∂η ⋆ 2+# (D9) The second term is proportional tos, and can be neglected for sufficiently smalls. We then have ∂F(s) ∂s s→0 ≈σ 2

    Linear stability analysis for the grand-canonical game We again carry out a linear stability analysis to obtain the stability criterion for the grand-canonical SK game. We find that fixed points are stable when σ2s * x⋆(1−x ⋆) λ−x ⋆(1−x ⋆)sΓσ2χ 2+ =σ 2s * ∂x⋆ ∂η ⋆ 2+ <1.(D7) As shown in Fig. 10, when Γ<0, increasing the fraction of speculators can destabi...

  13. [13]

    von Neumann and O

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

    1944

  14. [14]

    Hauert and G

    C. Hauert and G. Szab´ o, American Journal of Physics73, 405 (2005)

    2005

  15. [15]

    Traulsen and C

    A. Traulsen and C. Hauert, Reviews of Nonlinear Dynamics and Complexity2, 25 (2009)

    2009

  16. [16]

    J. F. Nash, Proceedings of the National Academy of Sciences36, 48 (1950)

    1950

  17. [17]

    Berg and A

    J. Berg and A. Engel, Phys. Rev. Lett.81, 4999 (1998)

    1998

  18. [18]

    Berg and M

    J. Berg and M. Weigt, Europhysics Letters48, 129 (1999)

    1999

  19. [19]

    McLennan and J

    A. McLennan and J. Berg, Games and Economic Behavior51, 264 (2005), special Issue in Honor of Richard D. McKelvey

    2005

  20. [20]

    Daskalakis, P

    C. Daskalakis, P. W. Goldberg, and C. H. Papadimitriou, Commun. ACM52, 89–97 (2009)

    2009

  21. [21]

    Y. Sato, E. Akiyama, and J. D. Farmer, Proceedings of the National Academy of Sciences99, 4748 (2002)

    2002

  22. [22]

    Sato and J

    Y. Sato and J. P. Crutchfield, Phys. Rev. E67, 015206(R) (2003)

    2003

  23. [23]

    Y. Sato, E. Akiyama, and J. P. Crutchfield, Physica D: Nonlinear Phenomena210, 21 (2005)

    2005

  24. [24]

    Galla and J

    T. Galla and J. D. Farmer, Proceedings of the National Academy of Sciences110, 1232 (2013)

    2013

  25. [25]

    J. B. Sanders, J. D. Farmer, and T. Galla, Scientific Reports8, 4902 (2018)

    2018

  26. [26]

    Pangallo, T

    M. Pangallo, T. Heinrich, and J. Doyne Farmer, Science Advances5, eaat1328 (2019)

    2019

  27. [27]

    Opper and S

    M. Opper and S. Diederich, Physical Review Letters69, 1616 (1992)

    1992

  28. [28]

    Garnier-Brun, M

    J. Garnier-Brun, M. Benzaquen, and J.-P. Bouchaud, Physical Review X14, 021039 (2024)

    2024

  29. [29]

    Sherrington and S

    D. Sherrington and S. Kirkpatrick, Physical Review Letters35, 1792 (1975)

    1975

  30. [30]

    P. C. Menezes and D. Sherrington, Journal of Physics A: Mathematical and Theoretical46, 505004 (2013)

    2013

  31. [31]

    R. M. May, Nature238, 413 (1972)

    1972

  32. [32]

    Challet, A

    D. Challet, A. De Martino, M. Marsili, and I. Perez Castillo, Journal of Statistical Mechanics: Theory and Experiment2006, P03004 (2006)

    2006

  33. [33]

    Camerer and T.-H

    C. Camerer and T.-H. Hua, Econometrica67, 827 (1999)

    1999

  34. [34]

    Traulsen, T

    A. Traulsen, T. R¨ ohl, and H. G. Schuster, Phys. Rev. Lett.93, 028701 (2004)

    2004

  35. [35]

    De Dominicis, Phys

    C. De Dominicis, Phys. Rev. B18, 4913 (1978)

    1978

  36. [36]

    A. C. C. Coolen,The Mathematical Theory of Minority Games: Statistical Mechanics of Interacting Agents(Oxford University Press, Oxford UK, 2005)

    2005

  37. [37]

    Galla, arXiv preprint arXiv:2405.14289 (2024)

    T. Galla, arXiv preprint arXiv:2405.14289 (2024)

    work page arXiv 2024

  38. [38]

    Bunin, Phys

    G. Bunin, Phys. Rev. E95, 042414 (2017)

    2017

  39. [39]

    F. Roy, G. Biroli, G. Bunin, and C. Cammarota, Journal of Physics A: Mathematical and Theoretical52, 484001 (2019)

    2019

  40. [40]

    Sidhom and T

    L. Sidhom and T. Galla, Phys. Rev. E101, 032101 (2020)

    2020

  41. [41]

    Challet, M

    D. Challet, M. Marsili, and Y.-C. Zhang, Physica A: Statistical Mechanics and its Applications276, 284 (2000). 24

    2000

  42. [42]

    Mandelbrot et al., Journal of Business36, 394 (1963)

    B. Mandelbrot et al., Journal of Business36, 394 (1963)

    1963

  43. [43]

    R. N. Mantegna and H. E. Stanley,Introduction to econophysics: correlations and complexity in finance(Cambridge University Press, 1999)

    1999

  44. [44]

    Rieger, Journal of Physics A: Mathematical and General22, 3447 (1989)

    H. Rieger, Journal of Physics A: Mathematical and General22, 3447 (1989). [33]Data and codes:,https://github.com/Desmondccw/complex_dynamics_in_sk_game

    1989

  45. [45]

    O’Rourke, D

    S. O’Rourke, D. Renfrew, A. Soshnikov, and V. Vu, Journal of Statistical Physics160, 89–119 (2015)

    2015