Game dynamics and Nash equilibria

July 2014, 1(3): 537-553. doi: 10.3934/jdg.2014.1.537

Game dynamics and Nash equilibria

Yannick Viossat ^1,

1.	CEREMADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, F-75775 Paris, France

Received November 2012 Revised July 2013 Published July 2014

Abstract
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There are games with a unique Nash equilibrium but such that, for almost all initial conditions, all strategies in the support of this equilibrium are eliminated by the replicator dynamics and the best-reply dynamics.

Keywords: best-reply dynamics., Nash equilibrium, replicator dynamics.

Mathematics Subject Classification: Primary: 91A22; Secondary: 34A34, 34A6.

Citation: Yannick Viossat. Game dynamics and Nash equilibria. Journal of Dynamics and Games, 2014, 1 (3) : 537-553. doi: 10.3934/jdg.2014.1.537

References:

[1]	A. Gaunersdorfer and J. Hofbauer, Fictitious play, Shapley polygons, and the replicator equation, Games and Economic Behavior, 11 (1995), 279-303. doi: 10.1006/game.1995.1052.
[2]	I. Gilboa and A. Matsui, Social stability and equilibrium, Econometrica, 59 (1991), 859-867. doi: 10.2307/2938230.
[3]	S. Hart, Adaptive heuristics, Econometrica, 73 (2005), 1401-1430. doi: 10.1111/j.1468-0262.2005.00625.x.
[4]	J. Hofbauer and W. H. Sandholm, Survival of dominated strategies under evolutionary dynamics, Theoretical Economics, 6 (2011), 341-377. doi: 10.3982/TE771.
[5]	J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998. doi: 10.1017/CBO9781139173179.
[6]	J. Hofbauer, S. Sorin and Y. Viossat, Time average replicator and best reply dynamics, Mathematics of Operations Research, 34 (2009), 263-269. doi: 10.1287/moor.1080.0359.
[7]	M. J. M. Jansen, Regularity and stability of equilibrium points of bimatrix games, Mathematics of Operations Research, 6 (1981), 530-550. doi: 10.1287/moor.6.4.530.
[8]	A. Matsui, Best-response dynamics and socially stable strategies, Journal of Economic Theory, 57 (1992), 343-362. doi: 10.1016/0022-0531(92)90040-O.
[9]	D. Monderer and A. Sela, Fictitious-play and No-Cycling Condition, SFB 504 Discussion Paper 97-12, Universität Mannheim, 1997.
[10]	W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT Press, Cambridge, MA, 2010.
[11]	P. D. Taylor and L. Jonker, Evolutionary stable strategies and game dynamics, Mathematical Biosciences, 40 (1978), 145-156. doi: 10.1016/0025-5564(78)90077-9.
[12]	E. van Damme, Stability and Perfection of Nash Equilibria, Second edition, Springer-Verlag, New-York, 1991. doi: 10.1007/978-3-642-58242-4.
[13]	Y. Viossat, The replicator dynamics does not lead to correlated equilibria, Games and Economic Behavior, 59 (2007), 397-407. doi: 10.1016/j.geb.2006.09.001.
[14]	Y. Viossat, Evolutionary dynamics may eliminate all strategies used in correlated equilibria, Mathematical Social Sciences, 56 (2008), 27-43. doi: 10.1016/j.mathsocsci.2007.12.001.
[15]	Y. Viossat, Deterministic monotone dynamics and dominated strategies, preprint,, , ().
[16]	J. W. Weibull, Evolutionary Game Theory, MIT Press, Cambridge, MA, 1995.
[17]	E. C. Zeeman, Population dynamics from game theory, in Global Theory of Dynamical Systems (eds. A. Nitecki and C. Robinson), Lecture Notes in Mathematics, 819, Springer, New York, 1980, 471-497.

show all references

References:

[1]	A. Gaunersdorfer and J. Hofbauer, Fictitious play, Shapley polygons, and the replicator equation, Games and Economic Behavior, 11 (1995), 279-303. doi: 10.1006/game.1995.1052.
[2]	I. Gilboa and A. Matsui, Social stability and equilibrium, Econometrica, 59 (1991), 859-867. doi: 10.2307/2938230.
[3]	S. Hart, Adaptive heuristics, Econometrica, 73 (2005), 1401-1430. doi: 10.1111/j.1468-0262.2005.00625.x.
[4]	J. Hofbauer and W. H. Sandholm, Survival of dominated strategies under evolutionary dynamics, Theoretical Economics, 6 (2011), 341-377. doi: 10.3982/TE771.
[5]	J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998. doi: 10.1017/CBO9781139173179.
[6]	J. Hofbauer, S. Sorin and Y. Viossat, Time average replicator and best reply dynamics, Mathematics of Operations Research, 34 (2009), 263-269. doi: 10.1287/moor.1080.0359.
[7]	M. J. M. Jansen, Regularity and stability of equilibrium points of bimatrix games, Mathematics of Operations Research, 6 (1981), 530-550. doi: 10.1287/moor.6.4.530.
[8]	A. Matsui, Best-response dynamics and socially stable strategies, Journal of Economic Theory, 57 (1992), 343-362. doi: 10.1016/0022-0531(92)90040-O.
[9]	D. Monderer and A. Sela, Fictitious-play and No-Cycling Condition, SFB 504 Discussion Paper 97-12, Universität Mannheim, 1997.
[10]	W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT Press, Cambridge, MA, 2010.
[11]	P. D. Taylor and L. Jonker, Evolutionary stable strategies and game dynamics, Mathematical Biosciences, 40 (1978), 145-156. doi: 10.1016/0025-5564(78)90077-9.
[12]	E. van Damme, Stability and Perfection of Nash Equilibria, Second edition, Springer-Verlag, New-York, 1991. doi: 10.1007/978-3-642-58242-4.
[13]	Y. Viossat, The replicator dynamics does not lead to correlated equilibria, Games and Economic Behavior, 59 (2007), 397-407. doi: 10.1016/j.geb.2006.09.001.
[14]	Y. Viossat, Evolutionary dynamics may eliminate all strategies used in correlated equilibria, Mathematical Social Sciences, 56 (2008), 27-43. doi: 10.1016/j.mathsocsci.2007.12.001.
[15]	Y. Viossat, Deterministic monotone dynamics and dominated strategies, preprint,, , ().
[16]	J. W. Weibull, Evolutionary Game Theory, MIT Press, Cambridge, MA, 1995.
[17]	E. C. Zeeman, Population dynamics from game theory, in Global Theory of Dynamical Systems (eds. A. Nitecki and C. Robinson), Lecture Notes in Mathematics, 819, Springer, New York, 1980, 471-497.

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