American Institute of Mathematical Sciences

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March  2011, 4(1): 187-213. doi: 10.3934/krm.2011.4.187

On a continuous mixed strategies model for evolutionary game theory

Astridh Boccabella 1, , Roberto Natalini 2, and  Lorenzo Pareschi 3,

1. 

Dipartimento di Scienze di Base e Applicate per l’Ingegneria (SBAI), Università degli Studi “Sapienza” di Roma, Italy
2. 

Istituto per le Applicazioni del Calcolo “M. Picone”, CNR, c/o Dip. di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1; I-00133 Roma
3. 

Dipartimento di Matematica & CMCS, Università degli Studi di Ferrara, Italy

Received  October 2010 Revised  November 2010 Published  January 2011

We consider an integro-differential model for evolutionary game theory which describes the evolution of a population adopting mixed strategies. Using a reformulation based on the first moments of the solution, we prove some analytical properties of the model and global estimates. The asymptotic behavior and the stability of solutions in the case of two strategies is analyzed in details. Numerical schemes for two and three strategies which are able to capture the correct equilibrium states are also proposed together with several numerical examples.
Keywords: Replicator dynamics, Evolutionary Game Theory, Kinetic equations, Continuous mixed strategies, Numerical methods..
Mathematics Subject Classification: 35Q91, 65R20, 45G1.
Citation: Astridh Boccabella, Roberto Natalini, Lorenzo Pareschi. On a continuous mixed strategies model for evolutionary game theory. Kinetic & Related Models, 2011, 4 (1) : 187-213. doi: 10.3934/krm.2011.4.187
References:
[1]

P. Abrams, Modelling the adaptive dynamics of traits involved in inter- and intraspecific interactions: An assessment of three methods,, Ecol. Lett., 4 (2001), 166. doi: 10.1046/j.1461-0248.2001.00199.x. Google Scholar

[2]

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D. Challet, M. Marsili and Y-C. Zhang, "Minority Games: Interacting Agents in Financial Markets,", Oxford University Press, (2005). Google Scholar

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R. Cressman, Stability of the replicator equation with continuous strategy space,, Mathematical Social Sciences, 50 (2005), 127. doi: 10.1016/j.mathsocsci.2005.03.001. Google Scholar

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L. Desvillettes, P.-E. Jabin, S. Mischler and G. Raoul, On selection dynamics for continuous structured populations,, Commun. Math. Sci., 6 (2008), 729. Google Scholar

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D. Friedman, Towards evolutionary game models of financial markets,, Quantitative Finance 1, (2001). Google Scholar

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A. Galstyan, Continuous strategy replicator dynamics for multi-agent learning,, \arXiv{0904.4717v1}., (). Google Scholar

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S. Genieys, N. Bessonov and V. Volpert, Mathematical model of evolutionary branching,, Mathematical and Computer Modelling, 49 (2009), 2109. doi: 10.1016/j.mcm.2008.07.018. Google Scholar

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J. Henriksson, T. Lundh and B. Wennberg, A model of sympatric speciation through reinforcement,, Kinet. Relat. Models, 3 (2010), 143. doi: 10.3934/krm.2010.3.143. Google Scholar

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J. Hofbauer, J. Oechssler and F. Riedel, Brown-von Neumann-Nash dynamics: The continuous strategy case,, Games and Economic Behavior, 65 (2009), 406. doi: 10.1016/j.geb.2008.03.006. Google Scholar

[11]

J. Hofbauer and K. Sigmund, "Evolutionary Games and Population Dynamics,", Cambridge University Press, (1998). Google Scholar

[12]

T. W. L. Norman, Dynamically stable sets in infinite strategy spaces,, Games and Economic Behavior, 62 (2008), 610. doi: 10.1016/j.geb.2007.05.005. Google Scholar

[13]

J. Oechssler and F. Riedel, Evolutionary dynamics on infinite strategy spaces,, Econ. Theory, 17 (2001), 141. doi: 10.1007/PL00004092. Google Scholar

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M. Ruijgrok and T. W. Ruijgrok, Replicator dynamics with mutations for games with a continuous strategy space,, \arXiv{nlin/0505032v2}., (). Google Scholar

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

show all references

References:
[1]

P. Abrams, Modelling the adaptive dynamics of traits involved in inter- and intraspecific interactions: An assessment of three methods,, Ecol. Lett., 4 (2001), 166. doi: 10.1046/j.1461-0248.2001.00199.x. Google Scholar

[2]

I. Bomze, Dynamical aspects of evolutionary stability,, Mon. Math., 110 (1990), 189. doi: 10.1007/BF01301675. Google Scholar

[3]

D. Challet, M. Marsili and Y-C. Zhang, "Minority Games: Interacting Agents in Financial Markets,", Oxford University Press, (2005). Google Scholar

[4]

R. Cressman, Stability of the replicator equation with continuous strategy space,, Mathematical Social Sciences, 50 (2005), 127. doi: 10.1016/j.mathsocsci.2005.03.001. Google Scholar

[5]

L. Desvillettes, P.-E. Jabin, S. Mischler and G. Raoul, On selection dynamics for continuous structured populations,, Commun. Math. Sci., 6 (2008), 729. Google Scholar

[6]

D. Friedman, Towards evolutionary game models of financial markets,, Quantitative Finance 1, (2001). Google Scholar

[7]

A. Galstyan, Continuous strategy replicator dynamics for multi-agent learning,, \arXiv{0904.4717v1}., (). Google Scholar

[8]

S. Genieys, N. Bessonov and V. Volpert, Mathematical model of evolutionary branching,, Mathematical and Computer Modelling, 49 (2009), 2109. doi: 10.1016/j.mcm.2008.07.018. Google Scholar

[9]

J. Henriksson, T. Lundh and B. Wennberg, A model of sympatric speciation through reinforcement,, Kinet. Relat. Models, 3 (2010), 143. doi: 10.3934/krm.2010.3.143. Google Scholar

[10]

J. Hofbauer, J. Oechssler and F. Riedel, Brown-von Neumann-Nash dynamics: The continuous strategy case,, Games and Economic Behavior, 65 (2009), 406. doi: 10.1016/j.geb.2008.03.006. Google Scholar

[11]

J. Hofbauer and K. Sigmund, "Evolutionary Games and Population Dynamics,", Cambridge University Press, (1998). Google Scholar

[12]

T. W. L. Norman, Dynamically stable sets in infinite strategy spaces,, Games and Economic Behavior, 62 (2008), 610. doi: 10.1016/j.geb.2007.05.005. Google Scholar

[13]

J. Oechssler and F. Riedel, Evolutionary dynamics on infinite strategy spaces,, Econ. Theory, 17 (2001), 141. doi: 10.1007/PL00004092. Google Scholar

[14]

M. Ruijgrok and T. W. Ruijgrok, Replicator dynamics with mutations for games with a continuous strategy space,, \arXiv{nlin/0505032v2}., (). Google Scholar

[15]

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

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