Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T04:02:34.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic Stability and Disagreements between Dynamics

Published online by Cambridge University Press:  01 January 2022

Aydin Mohseni
Get access Rights & Permissions [Opens in a new window]

Abstract

The replicator dynamics and Moran process are the main deterministic and stochastic models of evolutionary game theory. The models are connected by a mean-field relationship—the former describes the expected behavior of the latter. However, there are conditions under which their predictions diverge. I demonstrate that the divergence between their predictions is a function of standard techniques used in their analysis and of differences in the idealizations involved in each. My analysis reveals problems for stochastic stability analysis in a broad class of games, demonstrates a novel domain of agreement between the dynamics, and indicates a broader moral for evolutionary modeling.

Type
Articles
Information
Philosophy of Science , Volume 86 , Issue 3 , July 2019 , pp. 497 - 521
Copyright
Copyright © The Philosophy of Science Association

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

To contact the author, please write to: University of California, Irvine, Irvine, CA 92697; e-mail: amohseni@uci.edu.

†

I am indebted to Simon Huttegger, Brian Skyrms, Cailin O’Connor, Hannah Rubin, Cole Williams, and two anonymous referees for invaluable feedback on earlier drafts. Special thanks to audiences at the Generalized Theory of Evolution conference in Düsseldorf and the Infinite Idealizations in Science conference at the Munich Center for Mathematical Philosophy and participants in the Social Dynamics seminar in Irvine for helpful discussions.

References

Benaïm, M., and Weibull, J.. 2003. “Deterministic Approximation of Stochastic Evolution in Games.” Econometrica 71 (3): 873–903..CrossRefGoogle Scholar
Benaïm, M., and Weibull, J.. 2009. “Mean-Field Approximation of Stochastic Population Processes in Games.” Technical Report 1979, Ecole Polytechnique, Centre National de la Recherche Scientifique.Google Scholar
Binmore, K., and Samuelson, L.. 1997. “Muddling Through: Noisy Equilibrium Selection.” Journal of Economic Theory 74 (2): 235–65..CrossRefGoogle Scholar
Binmore, K., Samuelson, L., and Vaughan, R.. 1995. “Musical Chairs: Modeling Noisy Evolution.” Games and Economic Behavior 11 (1): 1–35..CrossRefGoogle Scholar
Bowles, S., and Gintis, H.. 2011. A Cooperative Species: Human Reciprocity and Its Evolution. Princeton, NJ: Princeton University Press.Google Scholar
Boyd, R., and Richerson, P. J.. 1985. Culture and the Evolutionary Process. vol. 175. Chicago: University of Chicago Press.Google Scholar
Cressman, R. 2003. Evolutionary Dynamics and Extensive Form Games. Cambridge, MA: MIT Press.CrossRefGoogle Scholar
Cressman, R., and Tao, Y.. 2014. “The Replicator Equation and Other Game Dynamics.” Proceedings of the National Academy of Sciences 111 (S3): 10810–17.CrossRefGoogle ScholarPubMed
Drake, J. W., Charlesworth, B., Charlesworth, D., and Crow, J. F.. 1998. “Rates of Spontaneous Mutation.” Genetics 148 (4): 1667–86..CrossRefGoogle ScholarPubMed
Foster, D., and Young, P.. 1990. “Stochastic Evolutionary Game Dynamics.” Theoretical Population Biology 38 (2): 219–32..CrossRefGoogle Scholar
Fudenberg, D., and Imhof, L.. 2004. “Stochastic Evolution as a Generalized Moran Process.” Unpublished manuscript, Harvard University, Program for Evolutionary Dynamics.Google Scholar
Fudenberg, D., and Imhof, L.. 2006. “Imitation Processes with Small Mutations.” Journal of Economic Theory 131 (1): 251–62..CrossRefGoogle Scholar
Fudenberg, D., Nowak, M. A., Taylor, C., and Imhof, L. A.. 2006. “Evolutionary Game Dynamics in Finite Populations with Strong Selection and Weak Mutation.” Theoretical Population Biology 70 (3): 352–63..CrossRefGoogle ScholarPubMed
García, J., and Traulsen, A.. 2012. “The Structure of Mutations and the Evolution of Cooperation.” PLoS ONE 7 (4).CrossRefGoogle ScholarPubMed
Hamilton, W. D. 1967. “Extraordinary Sex Ratios.” Science 156 (3774): 477–88..CrossRefGoogle ScholarPubMed
Harper, M., and Fryer, D.. 2016. “Stationary Stability for Evolutionary Dynamics in Finite Populations.” Entropy 18 (9).CrossRefGoogle Scholar
Imhof, L. A., and Nowak, M. A.. 2006. “Evolutionary Game Dynamics in a Wright-Fisher Process.” Journal of Mathematical Biology 52 (5): 667–81..CrossRefGoogle Scholar
Karlin, S., and Taylor, H. E.. 2012. A First Course in Stochastic Processes. 2nd ed. New York: Academic Press.Google Scholar
Kumar, S., and Subramanian, S.. 2002. “Mutation Rates in Mammalian Genomes.” Proceedings of the National Academy of Sciences of the USA 99 (2): 803–8..CrossRefGoogle ScholarPubMed
Maynard Smith, J. 1974. “The Theory of Games and the Evolution of Animal Conflicts.” Journal of Theoretical Biology 47 (1): 209–21..CrossRefGoogle Scholar
Maynard Smith, J.. 1982. Evolution and the Theory of Games. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Mohseni, A. 2017. “The Limitations of Equilibrium Concepts in Evolutionary Games.” Unpublished manuscript, University of California, Irvine, Department of Logic and Philosophy of Science.Google Scholar
Moran, P. A. 1962. The Statistical Processes of Evolutionary Theory. Oxford: Clarendon.Google Scholar
Nowak, M. A. 2006. Evolutionary Dynamics: Exploring the Equations of Life. Cambridge, MA: Belknap.CrossRefGoogle Scholar
Nowak, M. A., Sasaki, A., Taylor, C., and Fudenberg, D.. 2004. “Emergence of Cooperation and Evolutionary Stability in Finite Populations.” Nature 428 (6983): 646–50..CrossRefGoogle ScholarPubMed
Ohtsuki, H., Bordalo, P., and Nowak, M. A.. 2007. “The One-Third Law of Evolutionary Dynamics. Journal of Theoretical Biology 249 (2): 289–95..CrossRefGoogle Scholar
Sandholm, W. H. 2007. “Simple Formulas for Stationary Distributions and Stochastically Stable States.” Games and Economic Behavior 1:154–62.Google Scholar
Sandholm, W. H.. 2009. Population Games and Evolutionary Dynamics. Cambridge, MA: MIT Press.Google Scholar
Sandholm, W. H.. 2010. “Orders of Limits for Stationary Distributions, Stochastic Dominance, and Stochastic Stability.” Theoretical Economics 5 (1): 1–26..Google Scholar
Sandholm, W. H.. 2012. “Stochastic Imitative Game Dynamics with Committed Agents.” Journal of Economic Theory 5:2056–71.Google Scholar
Skyrms, B. 1996. Evolution of the Social Contract. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Taylor, C., Fudenberg, D., Sasaki, A., and Nowak, M. A.. 2004. “Evolutionary Game Dynamics in Finite Populations.” Bulletin of Mathematical Biology 66 (6): 1621–44..CrossRefGoogle ScholarPubMed
Taylor, P. D., and Jonker, L. B.. 1978. “Evolutionarily Stable Strategies and Game Dynamics.” Mathematical Biosciences 40 (1–2): 145–56.CrossRefGoogle Scholar
Tenaillon, O., Barrick, J. E., Ribeck, N., Deatherage, D. E., Blanchard, J. L., Dasgupta, A., Wu, G. C., Wielgoss, S., Cruveiller, S., Médigue, C., Schneider, D., and Lenski, R. E.. 2016. “Tempo and Mode of Genome Evolution in a 50,000-Generation Experiment.” Nature 536 (7615): 165–70..CrossRefGoogle Scholar
Traulsen, A., and Hauert, C.. 2010. “Stochastic Evolutionary Game Dynamics.” In Reviews of Nonlinear Dynamics and Complexity, Vol. 2, ed. Schuster, Heinz Georg, 25–61. Weinheim: Wiley.CrossRefGoogle Scholar
Young, H. P. 1993. “The Evolution of Conventions.” Econometrica 61 (1): 57–84..CrossRefGoogle Scholar
Young, H. P.. 1998. Individual Strategy and Social Structure: An Evolutionary Theory of Institutions. Princeton, NJ: Princeton University Press.CrossRefGoogle Scholar
Young, H. P.. 2006. “The Diffusion of Innovations in Social Networks.” In The Economy as an Evolving Complex System, III, ed. Blume, Lawrence E. and Durlauf, Steven N., 267–82. Oxford: Oxford University Press.Google Scholar
Young, H. P.. 2015. “The Evolution of Social Norms.” Annual Review of Economics 7 (1): 359–87..CrossRefGoogle Scholar