American Institute of Mathematical Sciences

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

[2]

I. Bomze, Dynamical aspects of evolutionary stability, Mon. Math., 110 (1990), 189-206. 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), 127-147. 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), 729-747.  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), 2109-2115. 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), 143-163. 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), 406-429. 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), 610-627. 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), 141-162. 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, Cambridge, MA, 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), 166-175. doi: 10.1046/j.1461-0248.2001.00199.x.  Google Scholar

[2]

I. Bomze, Dynamical aspects of evolutionary stability, Mon. Math., 110 (1990), 189-206. 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), 127-147. 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), 729-747.  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), 2109-2115. 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), 143-163. 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), 406-429. 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), 610-627. 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), 141-162. 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, Cambridge, MA, 1995.  Google Scholar

[1]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, 2021, 14 (1) : 115-148. doi: 10.3934/krm.2020051
[2]

Mirosław Lachowicz, Andrea Quartarone, Tatiana V. Ryabukha. Stability of solutions of kinetic equations corresponding to the replicator dynamics. Kinetic & Related Models, 2014, 7 (1) : 109-119. doi: 10.3934/krm.2014.7.109
[3]

Giacomo Albi, Lorenzo Pareschi, Mattia Zanella. Opinion dynamics over complex networks: Kinetic modelling and numerical methods. Kinetic & Related Models, 2017, 10 (1) : 1-32. doi: 10.3934/krm.2017001
[4]

Anna Lisa Amadori, Astridh Boccabella, Roberto Natalini. A hyperbolic model of spatial evolutionary game theory. Communications on Pure & Applied Analysis, 2012, 11 (3) : 981-1002. doi: 10.3934/cpaa.2012.11.981
[5]

William H. Sandholm. Local stability of strict equilibria under evolutionary game dynamics. Journal of Dynamics & Games, 2014, 1 (3) : 485-495. doi: 10.3934/jdg.2014.1.485
[6]

Xiulan Lai, Xingfu Zou. Dynamics of evolutionary competition between budding and lytic viral release strategies. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1091-1113. doi: 10.3934/mbe.2014.11.1091
[7]

Eric Foxall. Boundary dynamics of the replicator equations for neutral models of cyclic dominance. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1061-1082. doi: 10.3934/dcdsb.2020153
[8]

Sylvain Sorin. Replicator dynamics: Old and new. Journal of Dynamics & Games, 2020, 7 (4) : 365-386. doi: 10.3934/jdg.2020028
[9]

King-Yeung Lam. Dirac-concentrations in an integro-pde model from evolutionary game theory. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 737-754. doi: 10.3934/dcdsb.2018205
[10]

Sébastien Guisset. Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations. Kinetic & Related Models, 2020, 13 (4) : 739-758. doi: 10.3934/krm.2020025
[11]

Carlos Escudero, Fabricio Macià, Raúl Toral, Juan J. L. Velázquez. Kinetic theory and numerical simulations of two-species coagulation. Kinetic & Related Models, 2014, 7 (2) : 253-290. doi: 10.3934/krm.2014.7.253
[12]

Weizhu Bao, Yongyong Cai. Mathematical theory and numerical methods for Bose-Einstein condensation. Kinetic & Related Models, 2013, 6 (1) : 1-135. doi: 10.3934/krm.2013.6.1
[13]

Daewa Kim, Annalisa Quaini. A kinetic theory approach to model pedestrian dynamics in bounded domains with obstacles. Kinetic & Related Models, 2019, 12 (6) : 1273-1296. doi: 10.3934/krm.2019049
[14]

Joseph A. Connolly, Neville J. Ford. Comparison of numerical methods for fractional differential equations. Communications on Pure & Applied Analysis, 2006, 5 (2) : 289-307. doi: 10.3934/cpaa.2006.5.289
[15]

Vassilis G. Papanicolaou, Kyriaki Vasilakopoulou. Similarity solutions of a multidimensional replicator dynamics integrodifferential equation. Journal of Dynamics & Games, 2016, 3 (1) : 51-74. doi: 10.3934/jdg.2016003
[16]

Saul Mendoza-Palacios, Onésimo Hernández-Lerma. Stability of the replicator dynamics for games in metric spaces. Journal of Dynamics & Games, 2017, 4 (4) : 319-333. doi: 10.3934/jdg.2017017
[17]

Xiaomeng Li, Qiang Xu, Ailing Zhu. Weak Galerkin mixed finite element methods for parabolic equations with memory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 513-531. doi: 10.3934/dcdss.2019034
[18]

Michael Herty, Giuseppe Visconti. Kinetic methods for inverse problems. Kinetic & Related Models, 2019, 12 (5) : 1109-1130. doi: 10.3934/krm.2019042
[19]

Pierre Cardaliaguet, Chloé Jimenez, Marc Quincampoix. Pure and Random strategies in differential game with incomplete informations. Journal of Dynamics & Games, 2014, 1 (3) : 363-375. doi: 10.3934/jdg.2014.1.363
[20]

Shalva Amiranashvili, Raimondas  Čiegis, Mindaugas Radziunas. Numerical methods for a class of generalized nonlinear Schrödinger equations. Kinetic & Related Models, 2015, 8 (2) : 215-234. doi: 10.3934/krm.2015.8.215

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (92)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences