Pith. sign in

REVIEW 2 major objections 4 minor 1 cited by

Risk sensitivity creates attracting invariant curves and Arnold tongues in game learning

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · grok-4.3

2026-06-30 03:51 UTC pith:GEFFD55S

load-bearing objection Risk sensitivity plus adaptive beliefs in MWU produces Arnold tongues and codim-2 resonances that the risk-neutral version lacks, mapped via two-parameter bifurcation analysis. the 2 major comments →

arxiv 2606.29967 v1 pith:GEFFD55S submitted 2026-06-29 nlin.CD math.DS

Risk-Sensitive Learning in Population Games under Extreme Events: Bifurcations and Chaotic Dynamics

Konstantinos Metaxas , Themistoklis P. Sapsis This is my paper
classification nlin.CD math.DS
keywords risk-sensitive learningpopulation gamesextreme eventsbifurcation analysischaotic attractorsArnold tonguesmultiplicative weights update
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper studies the nonlinear dynamics of risk-sensitive multiplicative weights update in population congestion games when extreme events affect one action. The population state is coupled to a belief variable for perceived risk that updates via adaptive expectations. Two-parameter bifurcation analysis identifies regions with multi-stability, invariant curves, periodic and chaotic attractors. Risk incorporation produces new phenomena including attracting invariant curves that form phase-locking Arnold tongues with similar dynamics inside them. Codimension-two resonances organize the centers, and Cesaro averages converge to equilibrium despite the complexity.

Core claim

The resulting two-dimensional system exhibits complex behavior, including multi-stability among fixed points, invariant curves, periodic and chaotic attractors. Despite this complexity, the attractors can be grouped into distinct families, while the Cesaro averages of the trajectories are shown to converge to the stationary equilibrium. The incorporation of risk associated with the extreme event leads to new dynamical phenomena: attracting invariant curves arise and give rise to phase-locking Arnold tongues, within which the dynamics is qualitatively similar. In this setting, codimension-two resonances are identified as organizing centers, both within individual tongues and along the bifurca

What carries the argument

Risk-sensitive variant of the Multiplicative Weights Update (MWU) coupled with a belief variable capturing the agents' perceived risk and updated through an adaptive expectation rule.

Load-bearing premise

The population state follows a risk-sensitive variant of the Multiplicative Weights Update coupled with a belief variable capturing the agents' perceived risk and updated through an adaptive expectation rule.

What would settle it

Numerical continuation or simulation in the two-parameter plane to check for the emergence of attracting invariant curves and associated Arnold tongues as risk parameters vary, or verification if codimension-two resonances organize the observed bifurcations.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Attractors group into families with qualitatively similar dynamics within Arnold tongues
  • Cesaro averages of trajectories converge to the stationary equilibrium
  • Codimension-two resonances serve as organizing centers within tongues and along fixed-point bifurcations
  • Chaotic attractors form and vanish via Feigenbaum cascades, boundary crises, interior and merging crises
  • Transient chaos and narrow periodic windows occur in the dynamics

Where Pith is reading between the lines

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

  • The model implies that risk awareness can introduce phase-locking behaviors that might be observable in real decision-making under uncertainty.
  • Even with chaotic trajectories, long-term averages matching equilibrium suggests practical robustness of equilibrium predictions.
  • Similar bifurcation structures may appear in other learning rules or game types when risk is modeled explicitly.
  • Experimental setups inducing extreme events could test for the presence of Arnold tongues in agent behaviors.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The manuscript analyzes a two-dimensional discrete-time dynamical system arising from a risk-sensitive variant of the Multiplicative Weights Update (MWU) rule in population congestion games, coupled with an adaptive-expectations belief variable that tracks perceived risk from extreme events. It performs a two-parameter bifurcation analysis in the risk-sensitivity and adaptation-rate plane, classifies equilibria from both game-theoretic and dynamical viewpoints, and reports multi-stability among fixed points, attracting invariant curves, Arnold tongues with phase-locking, periodic and chaotic attractors (including Feigenbaum cascades, boundary crises, interior crises, and transient chaos), while showing that Cesàro averages of trajectories converge to the stationary Nash equilibrium. The incorporation of risk is claimed to generate qualitatively new phenomena—invariant curves and codimension-two resonances as organizing centers—not present in the risk-neutral case.

Significance. If the numerical bifurcation results and convergence statements hold, the work establishes that risk sensitivity under extreme events organizes new families of attractors (invariant curves giving rise to Arnold tongues) whose internal dynamics remain qualitatively similar across tongues, with codim-2 resonances serving as organizing centers both inside tongues and along fixed-point bifurcation curves. The grouping of attractors into families and the robust convergence of Cesàro averages to equilibrium despite local complexity are noteworthy for learning dynamics; the explicit phase-portrait and basin analysis supplies concrete evidence that strengthens the contribution to nonlinear dynamics in games.

major comments (2)
  1. [§4] §4 (bifurcation analysis of the 2D map): the claim that codimension-two resonances are the sole organizing centers for both the fixed-point family and the Arnold tongues rests on numerical continuation; the manuscript does not supply an analytic normal-form reduction or unfolding that would confirm the resonances are structurally stable under small perturbations of the risk-sensitivity and adaptation-rate parameters.
  2. [§5] §5 (convergence of Cesàro averages): the statement that averages converge to the stationary equilibrium for all attractors is supported only by representative trajectories; no theorem or Lyapunov-function argument is given that would guarantee this convergence uniformly across the parameter plane, particularly near the crisis boundaries where transient chaos is reported.
minor comments (4)
  1. [Abstract / §2] The abstract and §2 introduce the risk-sensitive MWU without an explicit equation number for the update rule; adding a numbered display equation at first use would improve traceability.
  2. [Figure captions] Figure captions for the phase portraits and basins (e.g., Figs. 7–12) omit the precise numerical values of the risk-sensitivity and adaptation-rate parameters used; these should be stated explicitly or referenced to the corresponding bifurcation diagram panels.
  3. [§5] The term “forward or reverse boundary crises” is used without a brief parenthetical definition or citation to the standard literature on crisis bifurcations; a short clarification would aid readers outside the dynamical-systems community.
  4. [References] The reference list contains several preprints whose arXiv numbers are given but whose final journal versions (if they exist) are not cross-checked; updating these would improve completeness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important distinctions between numerical evidence and analytic confirmation. We address each major comment below and will revise the manuscript accordingly to clarify the scope of our claims.

read point-by-point responses

  1. Referee: [§4] §4 (bifurcation analysis of the 2D map): the claim that codimension-two resonances are the sole organizing centers for both the fixed-point family and the Arnold tongues rests on numerical continuation; the manuscript does not supply an analytic normal-form reduction or unfolding that would confirm the resonances are structurally stable under small perturbations of the risk-sensitivity and adaptation-rate parameters.

    Authors: We agree that the identification of codimension-two resonances as organizing centers relies on numerical two-parameter continuation and bifurcation diagrams. The manuscript does not provide an analytic normal-form reduction or unfolding. We will revise the relevant passages in §4 to explicitly qualify these conclusions as numerically supported and to remove or temper the phrasing that they are the 'sole' organizing centers without analytic verification. A rigorous normal-form analysis of this particular map is beyond the present scope and is noted as future work. revision: yes

  2. Referee: [§5] §5 (convergence of Cesàro averages): the statement that averages converge to the stationary equilibrium for all attractors is supported only by representative trajectories; no theorem or Lyapunov-function argument is given that would guarantee this convergence uniformly across the parameter plane, particularly near the crisis boundaries where transient chaos is reported.

    Authors: The referee is correct that convergence of Cesàro averages to the Nash equilibrium is shown via representative trajectories and extensive numerical sampling rather than a general theorem or Lyapunov argument. This includes trajectories near crisis boundaries. We will revise §5 to state clearly that the result is numerically observed across the explored parameter regions (including near crises) but is not accompanied by a uniform proof. We view the numerical evidence as robust yet acknowledge the lack of a rigorous guarantee and flag this as an open question for subsequent analysis. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines the risk-sensitive MWU variant and adaptive belief update directly from first principles in the model setup, then applies standard dynamical systems tools (two-parameter bifurcation analysis, identification of Arnold tongues and codim-2 resonances) to the resulting 2D map. No step reduces a claimed prediction or uniqueness result to a fitted parameter or self-citation by construction; the new attractors and chaotic phenomena are shown to emerge from the stated equations without circular reduction. Cesàro averages converging to equilibrium is a direct consequence of the map definition, not a renamed input.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Abstract-only review limits visibility; the model introduces risk sensitivity and adaptive belief as core components without independent evidence supplied.

free parameters (2)
  • risk sensitivity parameter
    Controlled parameter varied in the two-parameter bifurcation analysis of the learning rule.
  • belief adaptation rate
    Parameter controlling update speed of the perceived-risk belief variable.
axioms (2)
  • domain assumption Population state evolves according to risk-sensitive variant of Multiplicative Weights Update
    Core modeling choice stated in first paragraph of abstract.
  • domain assumption Belief variable updated via adaptive expectation rule
    Assumption on how agents form perceived risk, stated in abstract.

pith-pipeline@v0.9.1-grok · 5816 in / 1391 out tokens · 48334 ms · 2026-06-30T03:51:38.468369+00:00 · methodology

0 comments
read the original abstract

Inspired by nonequilibrium phenomena in game dynamics and behavioral evidence on the impact of extreme events on decision making, we investigate the nonlinear dynamics of a discrete-time multiagent learning rule in population congestion games under extreme events affecting one of the actions. The population state, following a risk-sensitive variant of the Multiplicative Weights Update (MWU), is coupled with a belief variable capturing the agents perceived risk and updated through an adaptive expectation rule. We perform a two-parameter bifurcation analysis with respect to the agents controlled parameters, identifying regions of qualitatively distinct behavior. Equilibria are studied first from both game-theoretic and dynamical perspectives. The resulting two-dimensional system exhibits complex behavior, including multi-stability among fixed points, invariant curves, periodic and chaotic attractors. Despite this complexity, the attractors can be grouped into distinct families, while the Ces\`aro averages of the trajectories are shown to converge to the stationary equilibrium. The incorporation of risk associated with the extreme event leads to new dynamical phenomena: attracting invariant curves arise and give rise to phase-locking Arnold tongues, within which the dynamics is qualitatively similar. In this setting, codimension-two resonances are identified as organizing centers, both within individual tongues and along the bifurcation curves associated with the fixed-point family. Chaotic attractors emerge and are destroyed through Feigenbaum cascades and forward or reverse boundary crises, with interior and merging crises also observed, along with transient chaos and narrow periodic windows. For each qualitatively distinct region, representative phase portraits and the associated basins of attraction are examined.

Figures

Figures reproduced from arXiv: 2606.29967 by Konstantinos Metaxas, Themistoklis P. Sapsis.

Figure 1
Figure 1. Figure 1: Bifurcation curves in the first region. Saddle-node bifurcations are denoted by SN, period-doubling [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a), (b) The various emerging attractors as [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Basins of attraction of the coexisting attractors with [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Coexisting attractors in the phase space. (a) [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) Basins of attraction with a = 17.54 and k = 0.865: the chaotic attractor of family IV and the equilibrium coexist. (b) Basins of attraction with a = 18.245 and k = 0.865: a high-period orbit of the I-IV-V family and the equilibrium coexist. In both cases the numerical labels correspond to the Roman-numeral family indices, defined in [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The various attractors of the II–5 family in the phase space. (a) The attractor is a period-five orbit. (b) The attractor is a period-ten orbit, born at the PD2 − 5 curve of [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Phase space (a)-(d) and associated basins of attraction (e)-(h) with [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Phase space (a)-(d) and associated basins of attraction (e)-(h) with [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Bifurcation curves in the second region. Saddle-node bifurcations are denoted by SN, period-doubling [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The various emerging attractors as k = 0.865 is held fixed and a is gradually increased. Each attractor’s birth or destruction corresponds to the crossing of a bifurcation curve of [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Coexisting II–7, I–9, II–16, and I–25 attractors in the phase space (a) and their basins of attraction in [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Phase space (a)-(d) and associated basins of attraction (e)-(h) with [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Bifurcation curves in the third region, divided into two subregions shown in (a) and (c). In (b), an [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: (a) The various emerging attractors as k = 0.3 is held fixed and a is gradually increased. Each attractor’s birth or destruction corresponds to the crossing of a bifurcation curve of [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Attractors as a varies and k = 0.865 for the different set of fixed parameters. 4 Summary and conclusions In this paper, we studied the nonlinear dynamics of a risk-sensitive variant of the Multiplicative Weights Update (MWU) for two-strategy population congestion games in the presence of extreme events affecting the first resource. Agents are assumed to learn and react to the perceived risk, captured by … view at source ↗
Figure 16
Figure 16. Figure 16: (a) Attractor of the system defined by the map [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

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

  1. Natural Invariant Measures for Chaotic Game Dynamics: Finding Order in Chaos

    math.DS 2026-07 conditional novelty 6.0

    Despite Li-Yorke chaos, multiplicative-weights learning in a two-strategy congestion game still has natural invariant measures that fix long-run averages of payoffs, social cost, and regret.

Reference graph

Works this paper leans on

87 extracted references · 2 canonical work pages · cited by 1 Pith paper

  1. [1]

    Evolutionary dynamics on any population structure.Nature, 544(7649):227–230, 2017

    Benjamin Allen, Gabor Lippner, Yu-Ting Chen, Babak Fotouhi, Naghmeh Momeni, Shing-Tung Yau, and Martin A Nowak. Evolutionary dynamics on any population structure.Nature, 544(7649):227–230, 2017

    2017

  2. [2]

    Joaqu’in Andaluz, A. A. Elsadany, and Gloria Jarne. Dynamic Cournot oligopoly game based on general isoelastic demand.Nonlinear Dynamics, 99(2):1053–1063, 2020

    2020

  3. [3]

    Learning in matrix games can be arbitrarily complex

    Gabriel P Andrade, Rafael Frongillo, and Georgios Piliouras. Learning in matrix games can be arbitrarily complex. InConference on Learning Theory, pages 159–185. PMLR, 2021

    2021

  4. [4]

    The multiplicative weights update method: a meta-algorithm and applications.Theory of computing, 8(1):121–164, 2012

    Sanjeev Arora, Elad Hazan, and Satyen Kale. The multiplicative weights update method: a meta-algorithm and applications.Theory of computing, 8(1):121–164, 2012

    2012

  5. [5]

    Finite regret and cycles with fixed step-size via alternating gradient descent-ascent

    James P Bailey, Gauthier Gidel, and Georgios Piliouras. Finite regret and cycles with fixed step-size via alternating gradient descent-ascent. InConference on Learning Theory, pages 391–407. PMLR, 2020

    2020

  6. [6]

    Multiplicative weights update in zero-sum games

    James P Bailey and Georgios Piliouras. Multiplicative weights update in zero-sum games. InProceedings of the 2018 ACM Conference on Economics and Computation, pages 321–338, 2018

    2018

  7. [7]

    Multi-agent learning in network zero-sum games is a hamiltonian system

    James P Bailey and Georgios Piliouras. Multi-agent learning in network zero-sum games is a hamiltonian system. InProceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems, pages 233–241, 2019

    2019

  8. [8]

    Follow-the-regularized-leader routes to chaos in routing games

    Jakub Bielawski, Thiparat Chotibut, Fryderyk Falniowski, Grzegorz Kosiorowski, Michał Misiurewicz, and Georgios Piliouras. Follow-the-regularized-leader routes to chaos in routing games. InInternational Conference on Machine Learning, pages 925–935. PMLR, 2021. 22

    2021

  9. [9]

    Memory loss can prevent chaos in games dynamics.Chaos: An Interdisciplinary Journal of Nonlinear Science, 34(1), 2024

    Jakub Bielawski, Thiparat Chotibut, Fryderyk Falniowski, Michał Misiurewicz, and Georgios Piliouras. Memory loss can prevent chaos in games dynamics.Chaos: An Interdisciplinary Journal of Nonlinear Science, 34(1), 2024

    2024

  10. [10]

    Het- erogeneity, reinforcement learning, and chaos in population games.Proceedings of the National Academy of Sciences, 122(25):e2319929121, 2025

    Jakub Bielawski, Thiparat Chotibut, Fryderyk Falniowski, Michał Misiurewicz, and Georgios Piliouras. Het- erogeneity, reinforcement learning, and chaos in population games.Proceedings of the National Academy of Sciences, 122(25):e2319929121, 2025

    2025

  11. [11]

    On the influence of memory on complex dynamics of evolutionary oligopoly models.Nonlinear Dynamics, 102(2):1097–1110, 2020

    Gian Italo Bischi, Fabio Lamantia, and Bruno Scardamaglia. On the influence of memory on complex dynamics of evolutionary oligopoly models.Nonlinear Dynamics, 102(2):1097–1110, 2020

    2020

  12. [12]

    From Darwin to Poincaré and von Neumann: Recurrence and cycles in evolutionary and algorithmic game theory

    Victor Boone and Georgios Piliouras. From Darwin to Poincaré and von Neumann: Recurrence and cycles in evolutionary and algorithmic game theory. InInternational Conference on Web and Internet Economics, pages 85–99. Springer, 2019

    2019

  13. [13]

    Learning through reinforcement and replicator dynamics.Journal of economic theory, 77(1):1–14, 1997

    Tilman Börgers and Rajiv Sarin. Learning through reinforcement and replicator dynamics.Journal of economic theory, 77(1):1–14, 1997

    1997

  14. [14]

    Experience-weighted attraction learning in normal form games.Econometrica, 67(4):827–874, 1999

    Colin Camerer and Teck Hua Ho. Experience-weighted attraction learning in normal form games.Econometrica, 67(4):827–874, 1999

    1999

  15. [15]

    Behavioral economics.Econometric Society Monographs, 42:181, 2006

    Colin F Camerer, George Loewenstein, and Matthew Rabin. Behavioral economics.Econometric Society Monographs, 42:181, 2006

    2006

  16. [16]

    Effect of chaotic agent dynamics on coevolution of cooperation and synchronization.Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(11), 2020

    Rohitashwa Chattopadhyay, Shubhadeep Sadhukhan, and Sagar Chakraborty. Effect of chaotic agent dynamics on coevolution of cooperation and synchronization.Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(11), 2020

    2020

  17. [17]

    Learning in markets: Greed leads to chaos but following the price is right.arXiv preprint arXiv:2103.08529, 2021

    Yun Kuen Cheung, Stefanos Leonardos, and Georgios Piliouras. Learning in markets: Greed leads to chaos but following the price is right.arXiv preprint arXiv:2103.08529, 2021

    work page arXiv 2021

  18. [18]

    Vortices instead of equilibria in minmax optimization: Chaos and butterfly effects of online learning in zero-sum games

    Yun Kuen Cheung and Georgios Piliouras. Vortices instead of equilibria in minmax optimization: Chaos and butterfly effects of online learning in zero-sum games. InConference on Learning Theory, pages 807–834. PMLR, 2019

    2019

  19. [19]

    Chaos, extremism and optimism: Volume analysis of learning in games.Advances in Neural Information Processing Systems, 33:9039–9049, 2020

    Yun Kuen Cheung and Georgios Piliouras. Chaos, extremism and optimism: Volume analysis of learning in games.Advances in Neural Information Processing Systems, 33:9039–9049, 2020

    2020

  20. [20]

    The route to chaos in routing games: When is price of anarchy too optimistic?Advances in Neural Information Processing Systems, 33:766–777, 2020

    Thiparat Chotibut, Fryderyk Falniowski, Michał Misiurewicz, and Georgios Piliouras. The route to chaos in routing games: When is price of anarchy too optimistic?Advances in Neural Information Processing Systems, 33:766–777, 2020

    2020

  21. [21]

    Family of chaotic maps from game theory.Dynamical Systems, 36(1):48–63, 2021

    Thiparat Chotibut, Fryderyk Falniowski, Michał Misiurewicz, and Georgios Piliouras. Family of chaotic maps from game theory.Dynamical Systems, 36(1):48–63, 2021

    2021

  22. [22]

    DynamicalSystems

    George Datseris. DynamicalSystems. jl: A Julia software library for chaos and nonlinear dynamics.Journal of Open Source Software, 3(23):598, 2018

    2018

  23. [23]

    Basin entropy: a new tool to analyze uncertainty in dynamical systems.Scientific reports, 6(1):31416, 2016

    Alvar Daza, Alexandre Wagemakers, Bertrand Georgeot, David Guéry-Odelin, and Miguel AF Sanjuán. Basin entropy: a new tool to analyze uncertainty in dynamical systems.Scientific reports, 6(1):31416, 2016

    2016

  24. [24]

    Alvar Daza, Alexandre Wagemakers, and Miguel A.F. Sanjuán. Classifying basins of attraction using the basin entropy.Chaos, Solitons & Fractals, 159:112112, 2022

    2022

  25. [25]

    Dhooge, W

    A. Dhooge, W. Govaerts, and Yu. A. Kuznetsov. MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs.ACM Trans. Math. Softw., 29(2):141–164, jun 2003

    2003

  26. [26]

    Evolutionary dynamics of cooperation in optional public goods games with loners’ punishment.Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(3), 2025

    Rui Ding, Xianjia Wang, Ji Quan, Shuting Ma, and Jinhua Zhao. Evolutionary dynamics of cooperation in optional public goods games with loners’ punishment.Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(3), 2025

    2025

  27. [27]

    A stochastic variant of replicator dynamics in zero-sum games and its invariant measures.Physica D: Nonlinear Phenomena, 456:133940, 2023

    Maximilian Engel and Georgios Piliouras. A stochastic variant of replicator dynamics in zero-sum games and its invariant measures.Physica D: Nonlinear Phenomena, 456:133940, 2023

    2023

  28. [28]

    Punishment in bipartite societies

    Sinan Feng, Genjiu Xu, Yu Chen, Chaoqian Wang, and Attila Szolnoki. Punishment in bipartite societies. Chaos: An Interdisciplinary Journal of Nonlinear Science, 36(2), 2026

    2026

  29. [29]

    Adaptive game playing using multiplicative weights.Games and Economic Behavior, 29(1-2):79–103, 1999

    Yoav Freund and Robert E Schapire. Adaptive game playing using multiplicative weights.Games and Economic Behavior, 29(1-2):79–103, 1999. 23

    1999

  30. [30]

    MIT press, 1998

    Drew Fudenberg and David K Levine.The theory of learning in games, volume 2. MIT press, 1998

    1998

  31. [31]

    Evolutionary game dynamics in finite populations with strong selection and weak mutation.Theoretical population biology, 70(3):352–363, 2006

    Drew Fudenberg, Martin A Nowak, Christine Taylor, and Lorens A Imhof. Evolutionary game dynamics in finite populations with strong selection and weak mutation.Theoretical population biology, 70(3):352–363, 2006

    2006

  32. [32]

    Complex dynamics in learning complicated games.Proceedings of the National Academy of Sciences, 110(4):1232–1236, 2013

    Tobias Galla and J Doyne Farmer. Complex dynamics in learning complicated games.Proceedings of the National Academy of Sciences, 110(4):1232–1236, 2013

    2013

  33. [33]

    Celso Grebogi, Edward Ott, and James A. Yorke. Crises, sudden changes in chaotic attractors, and transient chaos.Physica D: Nonlinear Phenomena, 7(1):181–200, 1983

    1983

  34. [34]

    The replicator dynamics of zero-sum games arise from a novel poisson algebra.Chaos, Solitons & Fractals, 153:111508, 2021

    Christopher Griffin. The replicator dynamics of zero-sum games arise from a novel poisson algebra.Chaos, Solitons & Fractals, 153:111508, 2021

    2021

  35. [35]

    A finite population destroys a traveling wave in spatial replicator dynamics.Chaos, Solitons & Fractals, 146:110847, 2021

    Christopher Griffin, Riley Mummah, and Russ DeForest. A finite population destroys a traveling wave in spatial replicator dynamics.Chaos, Solitons & Fractals, 146:110847, 2021

    2021

  36. [36]

    Learning in nonatomic games, Part I: Finite action spaces and population games.Journal of Dynamics and Games, 9(4):433–460, 2022

    Saeed Hadikhanloo, Rida Laraki, Panayotis Mertikopoulos, and Sylvain Sorin. Learning in nonatomic games, Part I: Finite action spaces and population games.Journal of Dynamics and Games, 9(4):433–460, 2022

    2022

  37. [37]

    Integrating static and dynamic game theory with complex networks: Enhancing strategy dynamics through adaptive update rules.Chaos, Solitons & Fractals, 192:116063, 2025

    Reza Hakhamanesh, Javad Mohammadzadeh, Hadi Gholami Khaibary, and Masoomeh Azimi. Integrating static and dynamic game theory with complex networks: Enhancing strategy dynamics through adaptive update rules.Chaos, Solitons & Fractals, 192:116063, 2025

    2025

  38. [38]

    The description–experience gap in risky choice.Trends in cognitive sciences, 13(12):517–523, 2009

    Ralph Hertwig and Ido Erev. The description–experience gap in risky choice.Trends in cognitive sciences, 13(12):517–523, 2009

    2009

  39. [39]

    Cambridge university press, 1998

    Josef Hofbauer and Karl Sigmund.Evolutionary games and population dynamics. Cambridge university press, 1998

    1998

  40. [40]

    Dynamics of the cobweb model with adaptive expectations and nonlinear supply and demand

    Cars H Hommes. Dynamics of the cobweb model with adaptive expectations and nonlinear supply and demand. Journal of Economic Behavior & Organization, 24(3):315–335, 1994

    1994

  41. [41]

    Prospect theory: An analysis of decision under risk

    Daniel Kahneman and Amos Tversky. Prospect theory: An analysis of decision under risk. InHandbook of the fundamentals of financial decision making: Part I, pages 99–127. World Scientific, 2013

    2013

  42. [42]

    Simple and complex adjustment dynamics in cournot duopoly models.Chaos, Solitons & Fractals, 7(12):2031–2048, 1996

    Michael Kopel. Simple and complex adjustment dynamics in cournot duopoly models.Chaos, Solitons & Fractals, 7(12):2031–2048, 1996

    2031

  43. [43]

    On the convergence of no-regret learning in selfish routing

    Walid Krichene, Benjamin Drighes, and Alexandre Bayen. On the convergence of no-regret learning in selfish routing. InInternational Conference on Machine Learning, pages 163–171. PMLR, 2014

    2014

  44. [44]

    Turbulent coherent structures and early life below the kolmogorov scale.Nature communications, 11(1):2192, 2020

    Madison S Krieger, Sam Sinai, and Martin A Nowak. Turbulent coherent structures and early life below the kolmogorov scale.Nature communications, 11(1):2192, 2020

    2020

  45. [45]

    Yu. A. Kuznetsov.Elements of Applied Bifurcation Theory. Springer Science & Business Media, 3 edition, 2004

    2004

  46. [46]

    Noise-induced stochastic nash equilibrium in evolutionary game dynamics.Chaos, Solitons & Fractals, 192:115970, 2025

    Cong Li, Tianjiao Feng, Xiudeng Zheng, Sabin Lessard, and Yi Tao. Noise-induced stochastic nash equilibrium in evolutionary game dynamics.Chaos, Solitons & Fractals, 192:115970, 2025

    2025

  47. [47]

    Replicator dynamics of public goods game with tax-based punishment.Chaos, Solitons & Fractals, 164:112747, 2022

    MingYuan Li, HongWei Kang, XingPing Sun, Yong Shen, and QingYi Chen. Replicator dynamics of public goods game with tax-based punishment.Chaos, Solitons & Fractals, 164:112747, 2022

    2022

  48. [48]

    Springer Science & Business Media, 1992

    Allan J Lichtenberg and Michael A Lieberman.Regular and chaotic dynamics. Springer Science & Business Media, 1992

    1992

  49. [49]

    Memory and target payoff enhance cooperation in evolutionary social dilemmas.Chaos: An Interdisciplinary Journal of Nonlinear Science, 34(8), 2024

    Xinle Lin, Jianhe Li, and Suohai Fan. Memory and target payoff enhance cooperation in evolutionary social dilemmas.Chaos: An Interdisciplinary Journal of Nonlinear Science, 34(8), 2024

    2024

  50. [50]

    Evolutionary dynamics of cooperation in the public goods game with pool exclusion strategies.Nonlinear Dynamics, 97(1):749–766, 2019

    Linjie Liu, Xiaojie Chen, and Matjaž Perc. Evolutionary dynamics of cooperation in the public goods game with pool exclusion strategies.Nonlinear Dynamics, 97(1):749–766, 2019

    2019

  51. [51]

    Matouk, A

    Ahmed E. Matouk, A. A. Elsadany, and Baogui Xin. Neimark–sacker bifurcation analysis and complex nonlinear dynamics in a heterogeneous quadropoly game with an isoelastic demand function.Nonlinear Dynamics, 89(4):2533–2552, 2017. 24

    2017

  52. [52]

    Cycles in adversarial regularized learning

    Panayotis Mertikopoulos, Christos Papadimitriou, and Georgios Piliouras. Cycles in adversarial regularized learning. InProceedings of the twenty-ninth annual ACM-SIAM symposium on discrete algorithms, pages 2703–2717. SIAM, 2018

    2018

  53. [53]

    Evolutionary multi-agent reinforcement learning in group social dilemmas.Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(2), 2025

    Brian Mintz and Feng Fu. Evolutionary multi-agent reinforcement learning in group social dilemmas.Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(2), 2025

    2025

  54. [54]

    Potential games.Games and economic behavior, 14(1):124–143, 1996

    Dov Monderer and Lloyd S Shapley. Potential games.Games and economic behavior, 14(1):124–143, 1996

    1996

  55. [55]

    Deciphering chaos in evolutionary games.Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(12), 2020

    Archan Mukhopadhyay and Sagar Chakraborty. Deciphering chaos in evolutionary games.Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(12), 2020

    2020

  56. [56]

    Chaos and the evolution of cooperation.Proceedings of the National Academy of Sciences, 90(11):5091–5094, 1993

    Martin Nowak and Karl Sigmund. Chaos and the evolution of cooperation.Proceedings of the National Academy of Sciences, 90(11):5091–5094, 1993

    1993

  57. [57]

    Cambridge University Press, 2 edition, 2002

    Edward Ott.Chaos in Dynamical Systems. Cambridge University Press, 2 edition, 2002

    2002

  58. [58]

    Multiplicative weights update with constant step-size in congestion games: Convergence, limit cycles and chaos.Advances in Neural Information Processing Systems, 30, 2017

    Gerasimos Palaiopanos, Ioannis Panageas, and Georgios Piliouras. Multiplicative weights update with constant step-size in congestion games: Convergence, limit cycles and chaos.Advances in Neural Information Processing Systems, 30, 2017

    2017

  59. [59]

    Weight of fitness deviation governs strict physical chaos in replicator dynamics.Chaos: An Interdisciplinary Journal of Nonlinear Science, 28(3), 2018

    Varun Pandit, Archan Mukhopadhyay, and Sagar Chakraborty. Weight of fitness deviation governs strict physical chaos in replicator dynamics.Chaos: An Interdisciplinary Journal of Nonlinear Science, 28(3), 2018

    2018

  60. [60]

    Towards a taxonomy of learning dynamics in 2×2 games.Games and Economic Behavior, 132:1–21, 2022

    Marco Pangallo, James BT Sanders, Tobias Galla, and J Doyne Farmer. Towards a taxonomy of learning dynamics in 2×2 games.Games and Economic Behavior, 132:1–21, 2022

    2022

  61. [61]

    Spiral organization of quasi-periodic shrimp-shaped domains in a discrete predator–prey system

    NC Pati. Spiral organization of quasi-periodic shrimp-shaped domains in a discrete predator–prey system. Chaos: An Interdisciplinary Journal of Nonlinear Science, 34(8), 2024

    2024

  62. [62]

    Underweighting of rare events in strategic games.Experimental Economics, pages 1–27, 2025

    Ori Plonsky, Peter Ayoub, and Yefim Roth. Underweighting of rare events in strategic games.Experimental Economics, pages 1–27, 2025

    2025

  63. [63]

    Learning in settings with partial feedback and the wavy recency effect of rare events.Cognitive Psychology, 93:18–43, 2017

    Ori Plonsky and Ido Erev. Learning in settings with partial feedback and the wavy recency effect of rare events.Cognitive Psychology, 93:18–43, 2017

    2017

  64. [64]

    The probability weighting function.Econometrica, pages 497–527, 1998

    Drazen Prelec. The probability weighting function.Econometrica, pages 497–527, 1998

    1998

  65. [65]

    Chaos in duopoly pricing.Chaos, Solitons & Fractals, 1(6):573–581, 1991

    Tönu Puu. Chaos in duopoly pricing.Chaos, Solitons & Fractals, 1(6):573–581, 1991

    1991

  66. [66]

    Andreu Puy, Alvar Daza, Alexandre Wagemakers, and Miguel A.F. Sanjuán. A test for fractal boundaries based on the basin entropy.Communications in Nonlinear Science and Numerical Simulation, 95:105588, 2021

    2021

  67. [67]

    A class of games possessing pure-strategy nash equilibria.International journal of game theory, 2(1):65–67, 1973

    Robert W Rosenthal. A class of games possessing pure-strategy nash equilibria.International journal of game theory, 2(1):65–67, 1973

    1973

  68. [68]

    The prevalence of chaotic dynamics in games with many players.Scientific reports, 8(1):4902, 2018

    James BT Sanders, J Doyne Farmer, and Tobias Galla. The prevalence of chaotic dynamics in games with many players.Scientific reports, 8(1):4902, 2018

    2018

  69. [69]

    MIT press, 2010

    William H Sandholm.Population games and evolutionary dynamics. MIT press, 2010

    2010

  70. [70]

    Stability and diversity in collective adaptation.Physica D: Nonlinear Phenomena, 210(1-2):21–57, 2005

    Yuzuru Sato, Eizo Akiyama, and James P Crutchfield. Stability and diversity in collective adaptation.Physica D: Nonlinear Phenomena, 210(1-2):21–57, 2005

    2005

  71. [71]

    Chaos in learning a simple two-person game.Proceedings of the National Academy of Sciences, 99(7):4748–4751, 2002

    Yuzuru Sato, Eizo Akiyama, and J Doyne Farmer. Chaos in learning a simple two-person game.Proceedings of the National Academy of Sciences, 99(7):4748–4751, 2002

    2002

  72. [72]

    Coupled replicator equations for the dynamics of learning in multiagent systems.Physical Review E, 67(1):015206, 2003

    Yuzuru Sato and James P Crutchfield. Coupled replicator equations for the dynamics of learning in multiagent systems.Physical Review E, 67(1):015206, 2003

    2003

  73. [73]

    Fictitious play in 3×3 games: The transition between periodic and chaotic behaviour.Games and Economic Behavior, 63(1):259–291, 2008

    Colin Sparrow, Sebastian van Strien, and Christopher Harris. Fictitious play in 3×3 games: The transition between periodic and chaotic behaviour.Games and Economic Behavior, 63(1):259–291, 2008

    2008

  74. [74]

    Attractors and basins of cascading networked evolutionary game under generalized asynchronous update.Chaos, Solitons & Fractals, 200:117002, 2025

    Ruichen Sun, Hao Zhang, Chengye Zou, Chuan Zhang, and Zhengqi Liu. Attractors and basins of cascading networked evolutionary game under generalized asynchronous update.Chaos, Solitons & Fractals, 200:117002, 2025

    2025

  75. [75]

    J. M. T. Thompson, H. B. Stewart, and Y. Ueda. Safe, explosive, and dangerous bifurcations in dissipative dynamical systems.Phys. Rev. E, 49:1019–1027, Feb 1994. 25

    1994

  76. [76]

    Heterogeneous triopoly game with isoelastic demand function.Nonlinear Dynamics, 68:187–193, 2012

    Fabio Tramontana and Abd Elalim Abdo Elsadany. Heterogeneous triopoly game with isoelastic demand function.Nonlinear Dynamics, 68:187–193, 2012

    2012

  77. [77]

    Advances in prospect theory: Cumulative representation of uncertainty

    Amos Tversky and Daniel Kahneman. Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and uncertainty, 5(4):297–323, 1992

    1992

  78. [78]

    Fictitious play in 3×3 games: Chaos and dithering behaviour.Games and Economic Behavior, 73(1):262–286, 2011

    Sebastian van Strien and Colin Sparrow. Fictitious play in 3×3 games: Chaos and dithering behaviour.Games and Economic Behavior, 73(1):262–286, 2011

    2011

  79. [79]

    Chaos and unpredictability in evolutionary dynamics in discrete time.Physical review letters, 107(3):038101, 2011

    Daniele Vilone, Alberto Robledo, and Angel Sánchez. Chaos and unpredictability in evolutionary dynamics in discrete time.Physical review letters, 107(3):038101, 2011

    2011

  80. [80]

    Chaos persists in large-scale multi-agent learning despite adaptive learning rates.arXiv preprint arXiv:2306.01032, 2023

    Emmanouil-Vasileios Vlatakis-Gkaragkounis, Lampros Flokas, and Georgios Piliouras. Chaos persists in large-scale multi-agent learning despite adaptive learning rates.arXiv preprint arXiv:2306.01032, 2023

    work page arXiv 2023

Showing first 80 references.