Skip to main content
SpringerLink
Log in

Stochastic Stability in Spatial Games

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We discuss similarities and differences between systems of interacting players maximizing their individual payoffs and particles minimizing their interaction energy. Long-run behavior of stochastic dynamics of spatial games with multiple Nash equilibria is analyzed. In particular, we construct an example of a spatial game with three strategies, where stochastic stability of Nash equilibria depends on the number of players and the kind of dynamics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. W.B. Arthur, S.N. Durlauf,and D.A. Lane,eds.The Economy as an Evolving Complex System II.(Addison–Wesley,Reading, MA,1997).

    Google Scholar 

  2. Econophysics bulletin on www.unifr.ch/econophysics.

  3. J. Weibull,Evolutionary Game Theory (MIT Press, Cambridge,MA,1997).

  4. J. Hofbauer and K. Sigmund,Evolutionary Games and Population Dynamics (Cambridge University Press, Cambridge,1998).

    Google Scholar 

  5. J. Hofbauer and K. Sigmund,Evolutionary Game Dynamics,Bulletin AMS 40:479 (2003).

  6. L.E. Blume,The statistical mechanics of strategic interaction,Games Econ.Behav. 5:387 (1993).

    Google Scholar 

  7. G. Ellison,Learning,local interaction,and coordination,Econometrica 61:1047 (1993).

    Google Scholar 

  8. H.P. Young,Individual Strategy and Social Structure:An Evolutionary Theory of Institu-tions (Princeton University Press, Princeton,1998).

  9. G. Ellison,Basins of attraction,long-run stochastic stability,and the speed of step-by-step evolution,Rev.Econ.Stud. 67:17 (2000).

    Google Scholar 

  10. R. Axelrod,The Evolution of Cooperation (Basic Books, New York,1984).

  11. M.A. Nowak and R.M. May,The spatial dilemmas of evolution,Int.J.Bifurcation Chaos 3:35 (1993).

    Google Scholar 

  12. M.A. Nowak, S. Bonhoeffer,and R.M. May,More spatial games,Int.J.Bifurcation Chaos 4:33–56(1994).

    Google Scholar 

  13. K. Lindgren and M.G. Nordahl M.G.,Evolutionary dynamics of spatial games,Phys-ica D 75:292 (1994).

  14. K. Brauchli, T. Killingback,and M. Doebeli,Evolution of cooperation in spatially struc-tured populations,J.Theor.Biol. 200:405 (1999).

    Google Scholar 

  15. G.Szabó, T. Antal, P. Szabóand M. Droz,Spatial evolutionary prisoner's dilemma game with three strategies and external constraints,Phys.Rev.E 62:1095 (2000).

  16. Ch. Hauert,Effects of space in 2 ×2 games,Int.J.Bifurcation Chaos 12:1531 (2002).

  17. J. Maynard Smith,The theory of games and the evolution of animal conflicts,J.Theor. Biol. 47:209 (1974).

    Google Scholar 

  18. J. Maynard Smith,Evolution and the Theory of Games (Cambridge, Cambridge Univer-sity Press,1982).

    Google Scholar 

  19. P.D. Taylor and L.B. Jonker,Evolutionarily stable strategy and game dynamics,Math. Biosci. 40:145 (1978).

    Google Scholar 

  20. J. Hofbauer, P. Schuster,and K. Sigmund,A note on evolutionarily stable strategies and game dynamics,J.Theor.Biol. 81:609 (1979).

    Google Scholar 

  21. E. Zeeman,Dynamics of the evolution of animal conflicts,J.Theor.Biol. 89:249 (1981).

    Google Scholar 

  22. D. Foster and P.H. Young,Stochastic evolutionary game dynamics,Theor.Popul.Biol. 38:219 (1990).

    Google Scholar 

  23. Y. Kim,Equilibrium selection in n-person coordination games,Games Econ.Behav. 15:203 (1996).

    Google Scholar 

  24. M. Broom, C. Cannings, and G.T. Vickers,Multi-player matrix games,Bull.Math.Biol. 59:931 (1997).

    Google Scholar 

  25. M. Bukowski and J. Miękisz,Evolutionary and asymptotic stability in multi-player games,Warsaw University preprint (2002),www.mimuw.edu.pl/ ∼miekisz/multi.ps.

  26. J. Harsányi and R. Selten,A General Theory of Equilibrium Selection in Games (MIT Press, Cambridge MA,1988).

    Google Scholar 

  27. D. Monderer and L.S. Shapley,Potential games,Games Econ.Behav. 14:124 (1996).

    Google Scholar 

  28. T. Maruta,On the relationship between risk-dominance and stochastic stability,Games Econ.Behav. 19:221 (1977).

    Google Scholar 

  29. M. Freidlin and A. Wentzell,On small random perturbations of dynamical systems,Rus-sian Math.Surveys 25:1 (1970).

    Google Scholar 

  30. M. Freidlin and A. Wentzell,Random Perturbations of Dynamical Systems (Springer Ver-lag, New York,1984).

    Google Scholar 

  31. B. Shubert,A fow-graph formula for the stationary distribution of a Markov chain, IEEE Trans.Syst.Man.Cybernet.5:565 (1975).

    Google Scholar 

  32. M. Kandori, G.J. Mailath,and R. Rob,Learning,mutation,and long-run equilibria in games,Econometrica 61:29 (1993).

    Google Scholar 

  33. P.H. Young,The evolution of conventions,Econometrica 61:57 (1993).

    Google Scholar 

  34. J. Miękisz,Statistical mechanics of evolutionary games,Warsaw University preprint (2003),www.mimuw.edu.pl/ ∼miekisz/statmech.ps.

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics, Warsaw University, ul, Banacha 2, 02-097 Warsaw, Poland; e-mail:

    Jacek Miękisz

Authors
  1. Jacek Miękisz
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Miękisz, J. Stochastic Stability in Spatial Games. Journal of Statistical Physics 117, 99–110 (2004). https://doi.org/10.1023/B:JOSS.0000044065.65866.bc

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:JOSS.0000044065.65866.bc

Search

Navigation