American Institute of Mathematical Sciences

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July  2014, 1(3): 485-495. doi: 10.3934/jdg.2014.1.485

Local stability of strict equilibria under evolutionary game dynamics

William H. Sandholm 1,

1. 

Department of Economics, University of Wisconsin, 1180 Observatory Drive, Madison, WI 53706, United States

Received  November 2012 Revised  July 2013 Published  July 2014

We consider the stability of strict equilibrium under deterministic evolutionary game dynamics. We show that if the correlation between strategies' growth rates and payoffs is positive and bounded away from zero in a neighborhood of a strict equilibrium, then this equilibrium is locally stable.
Keywords: Evolutionary game theory, strict equilibria., local stability.
Mathematics Subject Classification: Primary: 37C70, 91A22; Secondary: 37C10, 91A1.
Citation: William H. Sandholm. Local stability of strict equilibria under evolutionary game dynamics. Journal of Dynamics & Games, 2014, 1 (3) : 485-495. doi: 10.3934/jdg.2014.1.485
References:
[1]

G. W. Brown and J. von Neumann, Solutions of games by differential equations,, in Contributions to the Theory of Games I, (1950), 73.   Google Scholar

[2]

R. Cressman, Local stability of smooth selection dynamics for normal form games,, Mathematical Social Sciences, 34 (1997), 1.  doi: 10.1016/S0165-4896(97)00009-7.  Google Scholar

[3]

S. Demichelis and K. Ritzberger, From evolutionary to strategic stability,, Journal of Economic Theory, 113 (2003), 51.  doi: 10.1016/S0022-0531(03)00078-4.  Google Scholar

[4]

D. Friedman, Evolutionary games in economics,, Econometrica, 59 (1991), 637.  doi: 10.2307/2938222.  Google Scholar

[5]

J. Hofbauer, Stability for the Best Response Dynamics,, Unpublished manuscript, (1995).   Google Scholar

[6]

J. Hofbauer, From Nash and Brown to Maynard Smith: Equilibria, dynamics, and ESS,, Selection, 1 (2000), 81.   Google Scholar

[7]

J. Hofbauer and W. H. Sandholm, Stable games and their dynamics,, Journal of Economic Theory, 144 (2009), 1665.  doi: 10.1016/j.jet.2009.01.007.  Google Scholar

[8]

J. Hofbauer, P. Schuster and K. Sigmund, A note on evolutionarily stable strategies and game dynamics,, Journal of Theoretical Biology, 81 (1979), 609.  doi: 10.1016/0022-5193(79)90058-4.  Google Scholar

[9]

J. Hofbauer and K. Sigmund, Theory of Evolution and Dynamical Systems,, Cambridge University Press, ().   Google Scholar

[10]

E. Hopkins, A note on best response dynamics,, Games and Economic Behavior, 29 (1999), 138.  doi: 10.1006/game.1997.0636.  Google Scholar

[11]

R. Lahkar and W. H. Sandholm, The projection dynamic and the geometry of population games,, Games and Economic Behavior, 64 (2008), 565.  doi: 10.1016/j.geb.2008.02.002.  Google Scholar

[12]

J. Maynard Smith and G. R. Price, The logic of animal conflict,, Nature, 246 (1973), 15.   Google Scholar

[13]

J. H. Nachbar, 'Evolutionary' selection dynamics in games: Convergence and limit properties,, International Journal of Game Theory, 19 (1990), 59.  doi: 10.1007/BF01753708.  Google Scholar

[14]

L. Samuelson and J. Zhang, Evolutionary stability in asymmetric games,, Journal of Economic Theory, 57 (1992), 363.  doi: 10.1016/0022-0531(92)90041-F.  Google Scholar

[15]

W. H. Sandholm, Potential games with continuous player sets,, Journal of Economic Theory, 97 (2001), 81.  doi: 10.1006/jeth.2000.2696.  Google Scholar

[16]

W. H. Sandholm, Excess payoff dynamics and other well-behaved evolutionary dynamics,, Journal of Economic Theory, 124 (2005), 149.  doi: 10.1016/j.jet.2005.02.003.  Google Scholar

[17]

W. H. Sandholm, Local stability under evolutionary game dynamics,, Theoretical Economics, 5 (2010), 27.  doi: 10.3982/TE505.  Google Scholar

[18]

W. H. Sandholm, Pairwise comparison dynamics and evolutionary foundations for Nash equilibrium,, Games, 1 (2010), 3.  doi: 10.3390/g1010003.  Google Scholar

[19]

W. H. Sandholm, Population Games and Evolutionary Dynamics,, MIT Press, (2010).   Google Scholar

[20]

B. Skyrms, The Dynamics of Rational Deliberation,, Harvard University Press, (1990).   Google Scholar

[21]

M. J. Smith, The stability of a dynamic model of traffic assignment-an application of a method of Lyapunov,, Transportation Science, 18 (1984), 245.  doi: 10.1287/trsc.18.3.245.  Google Scholar

[22]

J. M. Swinkels, Adjustment dynamics and rational play in games,, Games and Economic Behavior, 5 (1993), 455.  doi: 10.1006/game.1993.1025.  Google Scholar

[23]

P. D. Taylor and L. Jonker, Evolutionarily stable strategies and game dynamics,, Mathematical Biosciences, 40 (1978), 145.  doi: 10.1016/0025-5564(78)90077-9.  Google Scholar

[24]

J. W. Weibull, Evolutionary Game Theory,, MIT Press, (1995).   Google Scholar

[25]

J. W. Weibull, The mass action interpretation. Excerpt from 'The work of John Nash in game theory: Nobel Seminar, December 8, 1994'., Journal of Economic Theory, 69 (1996), 165.   Google Scholar

[26]

E. C. Zeeman, Population dynamics from game theory,, in Global Theory of Dynamical Systems (eds. Z. Nitecki and C. Robinson) (Evanston, (1979), 472.   Google Scholar

show all references

References:
[1]

G. W. Brown and J. von Neumann, Solutions of games by differential equations,, in Contributions to the Theory of Games I, (1950), 73.   Google Scholar

[2]

R. Cressman, Local stability of smooth selection dynamics for normal form games,, Mathematical Social Sciences, 34 (1997), 1.  doi: 10.1016/S0165-4896(97)00009-7.  Google Scholar

[3]

S. Demichelis and K. Ritzberger, From evolutionary to strategic stability,, Journal of Economic Theory, 113 (2003), 51.  doi: 10.1016/S0022-0531(03)00078-4.  Google Scholar

[4]

D. Friedman, Evolutionary games in economics,, Econometrica, 59 (1991), 637.  doi: 10.2307/2938222.  Google Scholar

[5]

J. Hofbauer, Stability for the Best Response Dynamics,, Unpublished manuscript, (1995).   Google Scholar

[6]

J. Hofbauer, From Nash and Brown to Maynard Smith: Equilibria, dynamics, and ESS,, Selection, 1 (2000), 81.   Google Scholar

[7]

J. Hofbauer and W. H. Sandholm, Stable games and their dynamics,, Journal of Economic Theory, 144 (2009), 1665.  doi: 10.1016/j.jet.2009.01.007.  Google Scholar

[8]

J. Hofbauer, P. Schuster and K. Sigmund, A note on evolutionarily stable strategies and game dynamics,, Journal of Theoretical Biology, 81 (1979), 609.  doi: 10.1016/0022-5193(79)90058-4.  Google Scholar

[9]

J. Hofbauer and K. Sigmund, Theory of Evolution and Dynamical Systems,, Cambridge University Press, ().   Google Scholar

[10]

E. Hopkins, A note on best response dynamics,, Games and Economic Behavior, 29 (1999), 138.  doi: 10.1006/game.1997.0636.  Google Scholar

[11]

R. Lahkar and W. H. Sandholm, The projection dynamic and the geometry of population games,, Games and Economic Behavior, 64 (2008), 565.  doi: 10.1016/j.geb.2008.02.002.  Google Scholar

[12]

J. Maynard Smith and G. R. Price, The logic of animal conflict,, Nature, 246 (1973), 15.   Google Scholar

[13]

J. H. Nachbar, 'Evolutionary' selection dynamics in games: Convergence and limit properties,, International Journal of Game Theory, 19 (1990), 59.  doi: 10.1007/BF01753708.  Google Scholar

[14]

L. Samuelson and J. Zhang, Evolutionary stability in asymmetric games,, Journal of Economic Theory, 57 (1992), 363.  doi: 10.1016/0022-0531(92)90041-F.  Google Scholar

[15]

W. H. Sandholm, Potential games with continuous player sets,, Journal of Economic Theory, 97 (2001), 81.  doi: 10.1006/jeth.2000.2696.  Google Scholar

[16]

W. H. Sandholm, Excess payoff dynamics and other well-behaved evolutionary dynamics,, Journal of Economic Theory, 124 (2005), 149.  doi: 10.1016/j.jet.2005.02.003.  Google Scholar

[17]

W. H. Sandholm, Local stability under evolutionary game dynamics,, Theoretical Economics, 5 (2010), 27.  doi: 10.3982/TE505.  Google Scholar

[18]

W. H. Sandholm, Pairwise comparison dynamics and evolutionary foundations for Nash equilibrium,, Games, 1 (2010), 3.  doi: 10.3390/g1010003.  Google Scholar

[19]

W. H. Sandholm, Population Games and Evolutionary Dynamics,, MIT Press, (2010).   Google Scholar

[20]

B. Skyrms, The Dynamics of Rational Deliberation,, Harvard University Press, (1990).   Google Scholar

[21]

M. J. Smith, The stability of a dynamic model of traffic assignment-an application of a method of Lyapunov,, Transportation Science, 18 (1984), 245.  doi: 10.1287/trsc.18.3.245.  Google Scholar

[22]

J. M. Swinkels, Adjustment dynamics and rational play in games,, Games and Economic Behavior, 5 (1993), 455.  doi: 10.1006/game.1993.1025.  Google Scholar

[23]

P. D. Taylor and L. Jonker, Evolutionarily stable strategies and game dynamics,, Mathematical Biosciences, 40 (1978), 145.  doi: 10.1016/0025-5564(78)90077-9.  Google Scholar

[24]

J. W. Weibull, Evolutionary Game Theory,, MIT Press, (1995).   Google Scholar

[25]

J. W. Weibull, The mass action interpretation. Excerpt from 'The work of John Nash in game theory: Nobel Seminar, December 8, 1994'., Journal of Economic Theory, 69 (1996), 165.   Google Scholar

[26]

E. C. Zeeman, Population dynamics from game theory,, in Global Theory of Dynamical Systems (eds. Z. Nitecki and C. Robinson) (Evanston, (1979), 472.   Google Scholar

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