Local stability of strict equilibria under evolutionary game dynamics

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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

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

Received November 2012 Revised July 2013 Published July 2014

Abstract
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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.
[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.
[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.
[4]	D. Friedman, Evolutionary games in economics,, Econometrica, 59 (1991), 637. doi: 10.2307/2938222.
[5]	J. Hofbauer, Stability for the Best Response Dynamics,, Unpublished manuscript, (1995).
[6]	J. Hofbauer, From Nash and Brown to Maynard Smith: Equilibria, dynamics, and ESS,, Selection, 1 (2000), 81.
[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.
[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.
[9]	J. Hofbauer and K. Sigmund, Theory of Evolution and Dynamical Systems,, Cambridge University Press, ().
[10]	E. Hopkins, A note on best response dynamics,, Games and Economic Behavior, 29 (1999), 138. doi: 10.1006/game.1997.0636.
[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.
[12]	J. Maynard Smith and G. R. Price, The logic of animal conflict,, Nature, 246 (1973), 15.
[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.
[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.
[15]	W. H. Sandholm, Potential games with continuous player sets,, Journal of Economic Theory, 97 (2001), 81. doi: 10.1006/jeth.2000.2696.
[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.
[17]	W. H. Sandholm, Local stability under evolutionary game dynamics,, Theoretical Economics, 5 (2010), 27. doi: 10.3982/TE505.
[18]	W. H. Sandholm, Pairwise comparison dynamics and evolutionary foundations for Nash equilibrium,, Games, 1 (2010), 3. doi: 10.3390/g1010003.
[19]	W. H. Sandholm, Population Games and Evolutionary Dynamics,, MIT Press, (2010).
[20]	B. Skyrms, The Dynamics of Rational Deliberation,, Harvard University Press, (1990).
[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.
[22]	J. M. Swinkels, Adjustment dynamics and rational play in games,, Games and Economic Behavior, 5 (1993), 455. doi: 10.1006/game.1993.1025.
[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.
[24]	J. W. Weibull, Evolutionary Game Theory,, MIT Press, (1995).
[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.
[26]	E. C. Zeeman, Population dynamics from game theory,, in Global Theory of Dynamical Systems* (eds. Z. Nitecki and C. Robinson) (Evanston*, (1979), 472.

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.
[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.
[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.
[4]	D. Friedman, Evolutionary games in economics,, Econometrica, 59 (1991), 637. doi: 10.2307/2938222.
[5]	J. Hofbauer, Stability for the Best Response Dynamics,, Unpublished manuscript, (1995).
[6]	J. Hofbauer, From Nash and Brown to Maynard Smith: Equilibria, dynamics, and ESS,, Selection, 1 (2000), 81.
[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.
[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.
[9]	J. Hofbauer and K. Sigmund, Theory of Evolution and Dynamical Systems,, Cambridge University Press, ().
[10]	E. Hopkins, A note on best response dynamics,, Games and Economic Behavior, 29 (1999), 138. doi: 10.1006/game.1997.0636.
[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.
[12]	J. Maynard Smith and G. R. Price, The logic of animal conflict,, Nature, 246 (1973), 15.
[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.
[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.
[15]	W. H. Sandholm, Potential games with continuous player sets,, Journal of Economic Theory, 97 (2001), 81. doi: 10.1006/jeth.2000.2696.
[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.
[17]	W. H. Sandholm, Local stability under evolutionary game dynamics,, Theoretical Economics, 5 (2010), 27. doi: 10.3982/TE505.
[18]	W. H. Sandholm, Pairwise comparison dynamics and evolutionary foundations for Nash equilibrium,, Games, 1 (2010), 3. doi: 10.3390/g1010003.
[19]	W. H. Sandholm, Population Games and Evolutionary Dynamics,, MIT Press, (2010).
[20]	B. Skyrms, The Dynamics of Rational Deliberation,, Harvard University Press, (1990).
[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.
[22]	J. M. Swinkels, Adjustment dynamics and rational play in games,, Games and Economic Behavior, 5 (1993), 455. doi: 10.1006/game.1993.1025.
[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.
[24]	J. W. Weibull, Evolutionary Game Theory,, MIT Press, (1995).
[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.
[26]	E. C. Zeeman, Population dynamics from game theory,, in Global Theory of Dynamical Systems* (eds. Z. Nitecki and C. Robinson) (Evanston*, (1979), 472.

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