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

On a continuous mixed strategies model for evolutionary game theory

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.
Citation: Astridh Boccabella, Roberto Natalini, Lorenzo Pareschi. On a continuous mixed strategies model for evolutionary game theory. Kinetic and 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), 166-175. doi: 10.1046/j.1461-0248.2001.00199.x.

[2]

I. Bomze, Dynamical aspects of evolutionary stability, Mon. Math., 110 (1990), 189-206. doi: 10.1007/BF01301675.

[3]

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

[4]

R. Cressman, Stability of the replicator equation with continuous strategy space, Mathematical Social Sciences, 50 (2005), 127-147. doi: 10.1016/j.mathsocsci.2005.03.001.

[5]

L. Desvillettes, P.-E. Jabin, S. Mischler and G. Raoul, On selection dynamics for continuous structured populations, Commun. Math. Sci., 6 (2008), 729-747.

[6]

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

[7]

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

[8]

S. Genieys, N. Bessonov and V. Volpert, Mathematical model of evolutionary branching, Mathematical and Computer Modelling, 49 (2009), 2109-2115. doi: 10.1016/j.mcm.2008.07.018.

[9]

J. Henriksson, T. Lundh and B. Wennberg, A model of sympatric speciation through reinforcement, Kinet. Relat. Models, 3 (2010), 143-163. doi: 10.3934/krm.2010.3.143.

[10]

J. Hofbauer, J. Oechssler and F. Riedel, Brown-von Neumann-Nash dynamics: The continuous strategy case, Games and Economic Behavior, 65 (2009), 406-429. doi: 10.1016/j.geb.2008.03.006.

[11]

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

[12]

T. W. L. Norman, Dynamically stable sets in infinite strategy spaces, Games and Economic Behavior, 62 (2008), 610-627. doi: 10.1016/j.geb.2007.05.005.

[13]

J. Oechssler and F. Riedel, Evolutionary dynamics on infinite strategy spaces, Econ. Theory, 17 (2001), 141-162. doi: 10.1007/PL00004092.

[14]

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

[15]

J. W. Weibull, "Evolutionary Game Theory," MIT Press, Cambridge, MA, 1995.

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), 166-175. doi: 10.1046/j.1461-0248.2001.00199.x.

[2]

I. Bomze, Dynamical aspects of evolutionary stability, Mon. Math., 110 (1990), 189-206. doi: 10.1007/BF01301675.

[3]

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

[4]

R. Cressman, Stability of the replicator equation with continuous strategy space, Mathematical Social Sciences, 50 (2005), 127-147. doi: 10.1016/j.mathsocsci.2005.03.001.

[5]

L. Desvillettes, P.-E. Jabin, S. Mischler and G. Raoul, On selection dynamics for continuous structured populations, Commun. Math. Sci., 6 (2008), 729-747.

[6]

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

[7]

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

[8]

S. Genieys, N. Bessonov and V. Volpert, Mathematical model of evolutionary branching, Mathematical and Computer Modelling, 49 (2009), 2109-2115. doi: 10.1016/j.mcm.2008.07.018.

[9]

J. Henriksson, T. Lundh and B. Wennberg, A model of sympatric speciation through reinforcement, Kinet. Relat. Models, 3 (2010), 143-163. doi: 10.3934/krm.2010.3.143.

[10]

J. Hofbauer, J. Oechssler and F. Riedel, Brown-von Neumann-Nash dynamics: The continuous strategy case, Games and Economic Behavior, 65 (2009), 406-429. doi: 10.1016/j.geb.2008.03.006.

[11]

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

[12]

T. W. L. Norman, Dynamically stable sets in infinite strategy spaces, Games and Economic Behavior, 62 (2008), 610-627. doi: 10.1016/j.geb.2007.05.005.

[13]

J. Oechssler and F. Riedel, Evolutionary dynamics on infinite strategy spaces, Econ. Theory, 17 (2001), 141-162. doi: 10.1007/PL00004092.

[14]

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

[15]

J. W. Weibull, "Evolutionary Game Theory," MIT Press, Cambridge, MA, 1995.

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