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The Explanatory Relevance of Nash Equilibrium: One-Dimensional Chaos in Boundedly Rational Learning

Published online by Cambridge University Press:  01 January 2022

Elliott Wagner
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Abstract

Game theory is often used to explain behavior. Such explanations often proceed by demonstrating that the behavior in question is a Nash equilibrium. Agents are in Nash equilibrium if each agent’s strategy maximizes her payoff given her opponents’ strategies. Nash equilibriums are fundamentally static, but it is usually assumed that equilibriums will be the outcome of a dynamic process of learning or evolution. This article demonstrates that, even in the most simple setting, this need not be true. In two-strategy games with just a single equilibrium, a family of imitative learning dynamics does not lead to equilibrium.

Type
General Philosophy of Science
Information
Philosophy of Science , Volume 80 , Issue 5 , December 2013 , pp. 783 - 795
Copyright
Copyright © The Philosophy of Science Association

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Footnotes

†

This material is based on work supported by the National Science Foundation under grant EF 1038456. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

References

Arneodo, A., Coullet, P., and Tresser, C.. 1980. “Occurrence of Strange Attractors in Three-Dimensional Volterra Equations.” Physics Letters A 79:259–63.Google Scholar
Benaïm, Michel, and Weibull, Jörgen W.. 2003. “Deterministic Approximation of Stochastic Evolution in Games.” Econometrica 71:873–903.CrossRefGoogle Scholar
Bernheim, B. Douglas. 1984. “Rationalizable Strategic Behavior.” Econometrica 52:1007–28.CrossRefGoogle Scholar
Björnerstedt, J., and Weibull, Jörgen W.. 1996. “Nash Equilibrium and Evolution by Imitation.” In Rationality in Economics, ed. Arrow, K. and Colombatto, E.. New York: Macmillan.Google Scholar
Cabrales, A., and Sobel, J.. 1992. “On the Limit Points of Discrete Selection Dynamics.” Journal of Economic Theory 57:407–19.CrossRefGoogle Scholar
Cressman, Ross. 2003. Evolutionary Dynamics and Extensive Form Games. Cambridge, MA: MIT Press.CrossRefGoogle Scholar
Dekel, E., and Scotchmer, S.. 1992. “On the Evolution of Optimizing Behavior.” Journal of Economic Theory 57:392–406.CrossRefGoogle Scholar
Fisher, R. A. 1930. The Genetical Theory of Natural Selection. Oxford: Clarendon.CrossRefGoogle Scholar
Fudenberg, Drew, and Levine, David K.. 1998. The Theory of Learning in Games. Cambridge, MA: MIT Press.Google Scholar
Gibbons, Robert. 1992. Game Theory for Applied Economists. Princeton, NJ: Princeton University Press.Google Scholar
Hamilton, W. D. 1968. “Extraordinary Sex Ratios.” Science 156:477–88.Google Scholar
Helbing, D. 1992. “A Mathematical Model for Behavioral Changes by Pair Interactions.” In Economic Evolution and Demographic Change: Formal Models in the Social Sciences, ed. Haag, G., Mueller, U., and Troitzsch, K. G., 330–48. Berlin: Springer.Google Scholar
Hofbauer, J., and Weibull, Jörgen W.. 1996. “Evolutionary Selection against Dominated Strategies.” Journal of Economic Theory 71:558–73.CrossRefGoogle Scholar
Maynard Smith, John. 1982. Evolution and the Theory of Games. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Pearce, David G. 1984. “Rationalizable Strategic Behavior and the Problem of Perfection.” Econometrica 52:1029–50.CrossRefGoogle Scholar
Sandholm, William H. 2010. Population Games and Evolutionary Dynamics. Cambridge, MA: MIT Press.Google Scholar
Sato, Yuzuru, Akiyama, Eizo, and Farmer, J. Doyne. 2002. “Chaos in Learning a Simple Two-Person Game.” Proceedings of the National Academy of Sciences 99:4748–51.CrossRefGoogle ScholarPubMed
Schnabl, Wolfgang, Stadler, Peter F., Forst, Christian, and Schuster, Peter. 1991. “Full Characterization of a Strange Attractor: Chaotic Dynamics in Low-Dimensional Replicator Systems.” Physica D 48:65–90.Google Scholar
Skyrms, Brian. 1992. “Chaos and the Explanatory Significance of Equilibrium: Strange Attractors in Evolutionary Game Dynamics.” In PSA 1992: Proceedings of the Biennial Meeting of the Philosophy of Science Association, 374–94. East Lansing, MI: Philosophy of Science Association.Google Scholar
Sober, Elliott. 1983. “Equilibrium Explanation.” Philosophical Studies 43:201–10.CrossRefGoogle Scholar
Strogatz, Steven H. 1994. Nonlinear Dynamics and Chaos. Cambridge, MA: Perseus.Google Scholar
Wagner, Elliott O. 2012. “Deterministic Chaos and the Evolution of Meaning.” British Journal for the Philosophy of Science 63:547–75.CrossRefGoogle Scholar