Abstract
Two examples demonstrate the possibility of extremely complicated non-convergent behavior in evolutionary game dynamics. For the Taylor-Jonker flow, the stable orbits for three strategies were investigated by Zeeman. Chaos does not occur with three strategies. This papers presents numerical evidence that chaotic dynamics on a “strange attractor” does occur with four strategies. Thus phenomenon is closely related to known examples of complicated behavior in Lotka-Volterra ecological models.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arneodo, A., Coullet, P. and Tresser, C., 1980, “Occurrence of strange attractors in three dimensional Volterra equations’, Physics Letters 79, 259–263.
Arneodo, A., Coullet, P., Peyraud, J. and Tresser, C., 1982, “Strange attractors in Volterra equations for species in competition”, Journal of Mathematical Biology 14, 153–157.
Baumol, W.J. and Benhabib, J., 1989, “Chaos: significance, mechanism and economic applications”, Journal of Economic Perspectives 3, 77–105.
Binmore, K., 1987, “Modeling rational players I and II”, Economics and Philosophy 3, 179–214; 4, 9–55.
Bomze, I.M., 1986, “Non-cooperative 2-person games in biology: a classification”, International Journal of Game Theory 15, 31–59.
Brown, G.W., 1951, “Iterative solutions of games by fictitious play”, in Activity Analysis of Production and Allocation (Cowles Commission Monograph) New York: Wiley. 374–376.
Cournot, A., 1897, Researches into the Mathematical Principles of the Theory of Wealth (tr. from the French ed. of 1838) New York: Macmillan.
Crawford, V., 1974, “Learning the optimal strategy in zero sum games”, Econometrica 42, 885–891.
Crawford, V., 1985, “Learning behavior and mixed-strategy Nash equilibria”, Journal of Economic Behavior and Organization 6, 69–78.
Crawford, V., 1989, “Learning behavior and mixed-strategy Nash equilibria”, Journal of Economic Behavior and Organization 6, 69–78.
Crawford, V., 1989, “Learning and mixed strategy equilibria in evolutionary games”, Journal of Evolutionary Biology 140, 537–550.
Crawford, V., 1991, “An ‘evolutionary’ explanation of Van Huyck, Battalio and Beil's experimental results on coordination”, Games and Economic Behavior 3, 25–59.
Crawford, V. and Haller, H. forthcoming “Learning how to cooperate: optimal play in repeated coordination games”, Econometrica.
van, Damme, E., 1987, Stability and Perfection of Nash Equilibria, Berlin: Springer.
Eells, E., 1984, “Metatickles and the dynamics of deliberation”, Theory and Decision 17, 71–95.
Gardini, L., Lupini, R. and Messia, M.G., 1989, “Hopf bifurcation and transition to chaos in Lotka-Volterra equation”, Mathematical Biology 27, 259–272.
Gilpin, M.E., 1979, “Spiral chaos in a predator-Prey model”, The American Naturalist 13, 306–308.
Guckenheimer, J. and Holmes, P., 1983, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Berlin: Springer.
Harsanyi, J.C. and Selten, R., 1988, A General Theory of Equilibrium Selection in Games Cambridge,Mass.: MIT Press.
Hofbauer, J., 1981, “On the occurrence of limit cycles in the Volterra-Lotka equation”, Nonlinear Analysis 5, 1003–1007.
Hofbauer, J. and Sigmund, K., 1988, The Theory of Evolution and Dynamical Systems Cambridge: Cambridge University Press.
Huberman, B.A. and Hogg, T., 1988, “The behavior of computational ecologies”, in The Ecology of Computation B.A. Huberman, ed., Amsterdam: North Holland 77–115.
Jeffrey, R., 1988, “How to probabilize a Newcomb problem”, in Probability and Causality J. Fetzer, ed., Dordrecht: Reidel 241–251.
Jordan, J.S., 1991, “Bayesian learning in normal form games”, Games and Economic Behavior 3, 82–100.
Kalai, E. and Lehrer, E., 1990, “Rational learning leads to Nash equilibrium”, Discussion Paper No. 858 J. L. Kellogg Graduate School of Management, Northwestern University.
May, R.M. and Leonard, W.L., 1975, “Nonlinear aspects of competition between three species”, SIAM Journal of Applied Mathematics 29, 243–253.
Maynard Smith, J., 1982, Evolution and the Theory of Games Cambridge: Cambridge University Press.
Maynard Smith, J. and Price, G.R., 1973, “The logic of animal conflict”, Nature 146, 15–18.
Milgrom, P. and Roberts, J., 1990, “Rationalizability, learning and equilibrium in games with strategic complementarities”, Econometrica 6, 1255–1277.
Milgrom, P. and Roberts, J., 1991, “Adaptive and sophisticated learning in normal form games”, Games and Economic Behavior 3, 82–100.
Miyasawa, K., 1961, “On the convergence of a learning process in a 2 × 2 non-zero-sum two-person game”, Research Memorandum 33, Princeton University.
Moulin, H., 1984, “Dominance solvability and Cournot-stability”, Mathematical Social Sciences 7, 83–102.
Myerson, R.B., Pollock, G.B. and Swinkels, J.M., 1991, “Viscous population equilibria”, Games and Economic Behavior 3, 101–109.
Nachbar, J.H., 1990, “‘Evolutionary’ selection dynamics in games: convergence and limit properties”, International Journal of Game Theory 19, 59–89.
Press, J., Flannery, B., Teukolsky, S. and Vetterling, W., 1989, Numerical Recipes: The Art of Scientific Computing rev. ed. Cambridge University Press.
Rand, D., 1978, “Exotic phenomena in games and Duopoly models”, Journal of Mathematical Economics 5, 173–184.
Robinson, J., 1951, “An iterative method of solving a game”, Annals of Mathematics 54, 296–301.
Rossler, O., 1976, “Different types of chaos in two simple differential equations”, Zeitschrift fur Naturforschung 31a, 1664–1670.
Samuelson, L., 1988, “Evolutionary foundations for solution concepts for finite, two-player, normal-form games”, in Proceedings of the Second Conference on Theoretical Aspects of Reasoning about Knowledge M. Vardi, ed., Los Altos, California: Morgan Kaufmann. 211–226.
Samuelson, L., 1991, “Limit evolutionary stable strategies in two-player normal form games”, Games and Economic Behavior 3, 110–128.
Samuelson, L. and Zhang, J., 1990, “Evolutionary stability in asymmetric games”, Discussion Paper — University of Wisconsin.
Selten, R., 1975, “Reexamination of the perfectness concept of equilibrium in extensive games”, International Journal of Game Theory 4, 25–55.
Selten, R., 1991, “Evolution, learning and economic behavior”, Games and Economic Behavior 3, 3–24.
Shaffer, W.M., 1985, “Order and chaos in ecological systems”, Ecology 66, 93–106.
Shaffer, W.M. and Kot, M., 1986, “Differential systems in ecology and epidemiology”, in Chaos: An Introduction A.V. Holden, ed., Manchester: University of Manchester Press, 158–178.
Shapley, L., 1964, “Some topics in two person games”, in Advances in Game Theory M. Dresher, L.S. Shapley and A.W. Tucker, eds., Princeton: Princeton University Press 1–28.
Skyrms, B., 1988, “Deliberational dynamics and the foundations of Bayesian game theory”, In Epistemology [Philosophical Perspectives v.2] J.E. Tomberlin, ed., Northridge: Ridgeview 345–367.
Skyrms, B., 1989, “Correlated equilibria and the dynamics of rational deliberation” Erkenntnis 31, 347–364.
Skyrms, B., 1990, The Dynamics of Rational Deliberation, Cambridge, Mass: Harvard University Press.
Skyrms, B., 1990, “Ratifiability and the logic of decision”, in Midwest Studies in Philosophy XV: The Philosophy of the Human Sciences P.A. French et al., eds., Notre Dame: University of Notre Dame Press 44–56.
Skyrms, B., 1991, “Inductive deliberation, admissible acts and perfect equilibrium”, in Foundations of Decision Theory M. Bacharach and S. Hurley, eds., Oxford: Blackwells 220–241.
Skyrms, B., forthcoming “Adaptive and inductive deliberational dynamics”, in CECOIA 2. P. Bourgine, ed., Pergamon Press: Oxford.
Smale, S., 1976, “On the differential equations of species in competition”, Journal of Mathematical Biology 3, 5–7.
Spohn, W., 1982, “How to make sense of game theory”, in Studies in Contemporary Economics vol 2: Philosophy of Economics W. Stegmuller et al., eds., Heidelberg and New York: Springer Verlag.
Stanford, W., 1991, “Prestable strategies in discounted duopoly games”, Games and Economic Behavior 3, 129–144.
Takeuchi, Y. and Adachi, N., 1984, “Influence of predation on species coexistence in Volterra models”, Mathematical Biosciences 70, 65–90.
Taylor, P. and Jonker, L., 1978, “Evolutionarily stable strategies and game dynamics”, Mathematical Biosciences 40, 145–156.
Thorlund-Peterson, L., 1990, “Iterative computation of Cournot equilibrium”, Games and Economic Behavior 2, 61–75.
Vance, R.R., 1978, “Predation and resource partitioning in a one-predator-two prey model community”, American Naturalist 112, 441–448.
Vandermeer, J., 1991, “Contributions to the global analysis of 3-D Lotka-Volterra equations: dynamic boundedness and indirect interactions in the case of one predator and two prey”, Journal of Theoretical Biology 148, 545–561.
von, Neumann, J. and Morgenstern, O., 1947, Theory of Games and Economic Behavior Princeton: Princeton University Press.
Wolf, A., 1986, “Quantifying chaos with Lyapunov exponents”, in Chaos A. Holden, ed., Manchester: Manchester University Press 273–290.
Zeeman, E.C., 1980, “Population dynamics from game theory”, in Global Theory of Dynamical Systems (Lecture Notes in Mathematics (819) Z. Niteck and C. Robinson, eds., Berlin: Springer Verlag.
Author information
Authors and Affiliations
Additional information
I would like to thank John Doyle for raising the question of chaos in game dynamics and for pointing out the review article of Baumol and Benhabib (1989). Steve Frank called my attention to Smale (1976) and Vandermeer (1991). Thanks also to Brad Armendt, J.H. Nachbar, Stergios Skapardas, Eliott Sober and an anonymous referee for comments on an earlier version of this paper.
Rights and permissions
About this article
Cite this article
Skyrms, B. Chaos in game dynamics. J Logic Lang Inf 1, 111–130 (1992). https://doi.org/10.1007/BF00171693
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00171693