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Multiple-trial conflicts and stochastic evolutionary game dynamics

Part of: Game theory Markov processes

Published online by Cambridge University Press:  01 July 2016

Lorens A. Imhof
Lorens A. Imhof*
Affiliation:
Bonn University
*
∗ Postal address: Statistische Abteilung und Hausdorff-Zentrum für Mathematik, Universität Bonn, Adenauerallee 24-42, 53113 Bonn, Germany. Email address: limhof@uni-bonn.de
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Abstract

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We consider stochastic replicator processes for games that are composed of finitely many trials. Several general results on the relation between Nash equilibria and the long-run behaviour of the stochastic processes are proved. In particular, a sufficient condition is given for almost sure convergence to a state where everyone plays in every trial a strict Nash equilibrium. The results are applied to multiple-trial conflicts based on wars of attrition and on sperm competition games with fair raffles, respectively.

Keywords

Nash equilibriumrandom strategystochastic differential equationstochastic replicator dynamicswar of attrition

MSC classification

Primary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Secondary: 91A22: Evolutionary games 92D25: Population dynamics (general)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 4 , December 2008 , pp. 1174 - 1197
Copyright
Copyright © Applied Probability Trust 2008 

References

Abramowitz, M. and Stegun, I. A. (1965). Handbook of Mathematical Functions. Dover, New York.Google Scholar
Akin, E. (1980). Domination or equilibrium. Math. Biosci. 50, 239–250.CrossRefGoogle Scholar
Benaïm, M., Hofbauer, J. and Sandholm, W. H. (2008). Robust permanence and impermanence for stochastic replicator dynamics. J. Biol. Dyn. 2, 180–195.CrossRefGoogle ScholarPubMed
Benaïm, M., Schreiber, S. J. and Tarrès, P. (2004). Generalized urn models of evolutionary processes. Ann. Appl. Prob. 14, 1455–1478.CrossRefGoogle Scholar
Bishop, D. T. and Cannings, C. (1978). A generalized war of attrition. J. Theoret. Biol. 70, 85–124.CrossRefGoogle ScholarPubMed
Cabrales, A. (2000). Stochastic replicator dynamics. Internat. Econom. Rev. 41, 451–481.CrossRefGoogle Scholar
Cannings, C. and Whittaker, J. C. (1995). The finite horizon war of attrition. Games Econom. Behav. 11, 193–236.CrossRefGoogle Scholar
Cressman, R. (1992). The Stability Concept of Evolutionary Game Theory. Springer, Berlin.CrossRefGoogle Scholar
Cressman, R. (2003). Evolutionary Dynamics and Extensive Form Games. MIT Press, Cambridge, MA.CrossRefGoogle Scholar
Foster, D. and Young, P. (1990). Stochastic evolutionary game dynamics. Theoret. Pop. Biol. 38, 219–232. (Correction 51 (1997), 77–78.)CrossRefGoogle Scholar
Friedman, A. (1975). Stochastic Differential Equations and Applications, Vol. 1. Academic Press, New York.Google Scholar
Fryer, T., Cannings, C. and Vickers, G. T. (1999). Sperm competition. I. Basic model, ESS and dynamics. J. Theoret. Biol. 196, 81–100.CrossRefGoogle ScholarPubMed
Fudenberg, D. and Harris, C. (1992). Evolutionary dynamics with aggregate shocks. J. Econom. Theory 57, 420–441.CrossRefGoogle Scholar
Gichman, I. I. and Skorochod, A. W. (1971). Stochastische Differentialgleichungen. Akademie, Berlin.Google Scholar
Haigh, J. (1975). Game theory and evolution. Adv. Appl. Prob. 7, 8–11.CrossRefGoogle Scholar
Has'minskiı˘, R. Z. (1980). Stochastic Stability of Differential Equations. Sijthoff and Noordhoff, Germantown, MD.CrossRefGoogle Scholar
Hofbauer, J. and Imhof, L. A. (2007). Time averages, recurrence and transience in the stochastic replicator dynamics. Preprint, Bonn University.Google Scholar
Hofbauer, J. and Sigmund, K. (1998). Evolutionary Games and Population Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Hofbauer, J. and Sigmund, K. (2003). Evolutionary game dynamics. Bull. Amer. Math. Soc. (N.S.) 40, 479–519.CrossRefGoogle Scholar
Imhof, L. A. (2005). The long-run behavior of the stochastic replicator dynamics. Ann. Appl. Prob. 15, 1019–1045.CrossRefGoogle Scholar
Imhof, L. A., Fudenberg, D. and Nowak, M. A. (2005). Evolutionary cycles of cooperation and defection. Proc. Nat. Acad. Sci. USA 102, 10797–10800.CrossRefGoogle ScholarPubMed
Karlin, S. (1959). Mathematical Methods and Theory in Games, Programming, and Economics, Vol. 1. Addison-Wesley, Reading, MA.Google Scholar
Karlin, S. (1968). Total Positivity. Stanford University Press.Google Scholar
Khasminskii, R. and Potsepun, N. (2006). On the replicator dynamics behavior under Stratonovich type random perturbations. Stoch. Dyn. 6, 197–211.CrossRefGoogle Scholar
Maynard Smith, J. and Price, G. R. (1973). The logic of animal conflict. Nature 246, 15–18.CrossRefGoogle Scholar
Nowak, M. A. and Sigmund, K. (2004). Evolutionary dynamics of biological games. Science 303, 793–799.CrossRefGoogle ScholarPubMed
Parker, G. A. (1990). Sperm competition games: raffles and roles. Proc. R. Soc. London B 242, 120–126.Google Scholar
Samuelson, L. and Zhang, J. (1992). Evolutionary stability in asymmetric games. J. Econom. Theory 57, 363–391.CrossRefGoogle Scholar
Schreiber, S. J. (2001). Urn models, replicator processes, and random genetic drift. SIAM J. Appl. Math. 61, 2148–2167.CrossRefGoogle Scholar
Skorokhod, A. V. (1989). Asymptotic Methods in the Theory of Stochastic Differential Equations. American Mathematical Society, Providence, RI.Google Scholar
Taylor, P. D. and Jonker, L. B. (1978). Evolutionarily stable strategies and game dynamics. Math. Biosci. 40, 145–156.CrossRefGoogle Scholar
Weibull, J. W. (1995). Evolutionary Game Theory. MIT Press, Cambridge, MA.Google Scholar
Whittaker, J. C. (1996). The allocation of resources in a multiple-trial war of attrition conflict. Adv. Appl. Prob. 28, 933–964.CrossRefGoogle Scholar
Whittaker, J. C. and Cannings, C. (1994). A resource allocation problem. J. Theoret. Biol. 167, 397–405.CrossRefGoogle Scholar