previous ::
next
The long-run behavior of the stochastic replicator dynamics
Lorens A. Imhof
Source: Ann. Appl. Probab. Volume 15, Number 1B (2005), 1019-1045.
Abstract
Fudenberg and Harris’ stochastic version of the classical replicator dynamics is considered. The behavior of this diffusion process in the presence of an evolutionarily stable strategy is investigated. Moreover, extinction of dominated strategies and stochastic stability of strict Nash equilibria are studied. The general results are illustrated in connection with a discrete war of attrition. A persistence result for the maximum effort strategy is obtained and an explicit expression for the evolutionarily stable strategy is derived.
Primary Subjects: 60H10, 60J70, 92D15, 92D25
Keywords: Asymptotic stochastic stability; evolutionarily stable strategy; invariant measure; Lyapunov function; Nash equilibrium; recurrence; stochastic differential equation; war of attrition
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aoap/1107271677
Digital Object Identifier: doi:10.1214/105051604000000837
Mathematical Reviews number (MathSciNet):
MR2114999
Zentralblatt MATH identifier:
1081.60045
References
Akin, E. (1980). Domination or equilibrium. Math. Biosci. 50 239--250.
Mathematical Reviews (MathSciNet):
MR642308
Digital Object Identifier: doi:10.1016/0025-5564(80)90039-5
Alvarez, L. H. R. (2000). On the comparative static properties of the expected population density in the presence of stochastic fluctuations. J. Math. Biol. 40 432--442.
Mathematical Reviews (MathSciNet):
MR1761532
Digital Object Identifier: doi:10.1007/s002850000028
Amir, M. and Berninghaus, S. (1998). Scale functions in equilibrium selection games. Journal of Evolutionary Economics 8 1--13.
Bapat, R. B. and Raghavan, T. E. S. (1997). Nonnegative Matrices and Applications. Cambridge Univ. Press.
Mathematical Reviews (MathSciNet):
MR1449393
Zentralblatt MATH:
0879.15015
Bhattacharya, R. N. (1978). Criteria for recurrence and existence of invariant measures for multidimensional diffusions. Ann. Probab. 6 541--553.
Mathematical Reviews (MathSciNet):
MR494525
JSTOR: links.jstor.org
Bishop, D. T. and Cannings, C. (1978). A generalized war of attrition. J. Theoret. Biol. 70 85--124.
Mathematical Reviews (MathSciNet):
MR472121
Digital Object Identifier: doi:10.1016/0022-5193(78)90304-1
Cabrales, A. (2000). Stochastic replicator dynamics. International Economic Review 41 451--481.
Mathematical Reviews (MathSciNet):
MR1760461
Digital Object Identifier: doi:10.1111/1468-2354.00071
Corradi, V. and Sarin, R. (2000). Continuous approximations of stochastic evolutionary game dynamics. J. Econom. Theory 94 163--191.
Mathematical Reviews (MathSciNet):
MR1791297
Digital Object Identifier: doi:10.1006/jeth.1999.2596
Cressman, R. (2003). Evolutionary Dynamics and Extensive Form Games. MIT Press.
Mathematical Reviews (MathSciNet):
MR2004666
Zentralblatt MATH:
1067.91001
Durrett, R. (1996). Stochastic Calculus. CRC Press, Boca Raton, FL.
Mathematical Reviews (MathSciNet):
MR1398879
Zentralblatt MATH:
0856.60002
Foster, D. and Young, P. (1990). Stochastic evolutionary game dynamics. Theoret. Population Biol. 38 219--232. [Corrigendum Theoret. Population Biol. 51 (1997) 77--78.]
Mathematical Reviews (MathSciNet):
MR1075000
Digital Object Identifier: doi:10.1016/0040-5809(90)90011-J
Fudenberg, D. and Harris, C. (1992). Evolutionary dynamics with aggregate shocks. J. Econom. Theory 57 420--441.
Mathematical Reviews (MathSciNet):
MR1180005
Digital Object Identifier: doi:10.1016/0022-0531(92)90044-I
Gantmacher, F. R. (1959). The Theory of Matrices 1. Chelsea, New York.
Zentralblatt MATH:
0085.01001
Gichman, I. I. and Skorochod, A. W. (1971). Stochastische Differentialgleichungen. Akademie Verlag, Berlin.
Mathematical Reviews (MathSciNet):
MR346905
Zentralblatt MATH:
0287.34063
Haigh, J. (1975). Game theory and evolution. Adv. in Appl. Probab. 7 8--11.
Hofbauer, J. and Sigmund, K. (1998). Evolutionary Games and Population Dynamics. Cambridge Univ. Press.
Mathematical Reviews (MathSciNet):
MR1635735
Zentralblatt MATH:
0914.90287
Hofbauer, J. and Sigmund, K. (2003). Evolutionary game dynamics. Bull. Amer. Math. Soc. (N.S.) 40 479--519.
Mathematical Reviews (MathSciNet):
MR1997349
Digital Object Identifier: doi:10.1090/S0273-0979-03-00988-1
Kandori, M., Mailath, G. J. and Rob, R. (1993). Learning, mutation, and long run equilibria in games. Econometrica 61 29--56.
Mathematical Reviews (MathSciNet):
MR1201702
Khas'minskii, R. Z. (1960). Ergodic properties of recurrent diffusion processes and stabilization of the solution to the Cauchy problem for parabolic equations. Theory Probab. Appl. 5 179--196.
Mathematical Reviews (MathSciNet):
MR133871
Khasminskii, R. Z. and Klebaner, F. C. (2001). Long term behavior of solutions of the Lotka--Volterra system under small random perturbations. Ann. Appl. Probab. 11 952--963.
Mathematical Reviews (MathSciNet):
MR1865029
Digital Object Identifier: doi:10.1214/aoap/1015345354
Project Euclid: euclid.aoap/1015345354
Klebaner, F. C. and Liptser, R. (2001). Asymptotic analysis and extinction in a stochastic Lotka--Volterra model. Ann. Appl. Probab. 11 1263--1291.
Mathematical Reviews (MathSciNet):
MR1878298
Digital Object Identifier: doi:10.1214/aoap/1015345403
Project Euclid: euclid.aoap/1015345403
Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory and Majorization and Its Applications. Academic Press, New York.
Mathematical Reviews (MathSciNet):
MR552278
Zentralblatt MATH:
0437.26007
Maynard Smith, J. (1982). Evolution and the Theory of Games. Cambridge Univ. Press.
Maynard Smith, J. and Price, G. R. (1973). The logic of animal conflict. Nature 246 15--18.
Nowak, M. A., Sasaki, A., Taylor, C. and Fudenberg, D. (2004). Emergence of cooperation and evolutionary stability in finite populations. Nature 428 646--650.
Nowak, M. A. and Sigmund, K. (2004). Evolutionary dynamics of biological games. Science 303 793--799.
Saito, M. (1997). A note on ergodic distributions in two-agent economies. J. Math. Econom. 27 133--141.
Mathematical Reviews (MathSciNet):
MR1431151
Digital Object Identifier: doi:10.1016/S0304-4068(96)00775-6
Skorokhod, A. V., Hoppensteadt, F. C. and Salehi, H. (2002). Random Perturbation Methods with Applications in Science and Engineering. Springer, New York.
Mathematical Reviews (MathSciNet):
MR1912425
Zentralblatt MATH:
0998.37001
Szegö, G. (1975). Orthogonal Polynomials. Amer. Math. Soc., Providence, RI.
Mathematical Reviews (MathSciNet):
MR372517
Taylor, P. D. and Jonker, L. B. (1978). Evolutionarily stable strategies and game dynamics. Math. Biosci. 40 145--156.
Mathematical Reviews (MathSciNet):
MR489983
Digital Object Identifier: doi:10.1016/0025-5564(78)90077-9
Weibull, J. W. (1995). Evolutionary Game Theory. MIT Press, Cambridge, MA.
Mathematical Reviews (MathSciNet):
MR1347921
Zentralblatt MATH:
0879.90206
Whittaker, J. C. (1996). The allocation of resources in a multiple trial war of attrition conflict. Adv. in Appl. Probab. 28 933--964.
Mathematical Reviews (MathSciNet):
MR1404316
previous ::
next
The Annals of Applied Probability
