On a continuous mixed strategies model for evolutionary game theory

Previous Article
Kinetic modeling of economic games with large number of participants
KRM Home
This Issue
Next Article
On a charge interacting with a plasma of unbounded mass

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

Abstract
Related Papers

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.
[2]	I. Bomze, Dynamical aspects of evolutionary stability,, Mon. Math., 110 (1990), 189. 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. 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.
[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. 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. 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. 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. 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. 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, (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. doi: 10.1046/j.1461-0248.2001.00199.x.
[2]	I. Bomze, Dynamical aspects of evolutionary stability,, Mon. Math., 110 (1990), 189. 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. 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.
[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. 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. 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. 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. 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. 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, (1995).

[1]	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
[2]	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
[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]	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
[5]	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
[6]	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
[7]	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
[8]	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
[9]	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
[10]	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
[11]	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
[12]	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
[13]	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
[14]	Yannick Viossat. Game dynamics and Nash equilibria. Journal of Dynamics & Games, 2014, 1 (3) : 537-553. doi: 10.3934/jdg.2014.1.537
[15]	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
[16]	Ya-Xiang Yuan. Recent advances in numerical methods for nonlinear equations and nonlinear least squares. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 15-34. doi: 10.3934/naco.2011.1.15
[17]	Iasson Karafyllis, Lars Grüne. Feedback stabilization methods for the numerical solution of ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 283-317. doi: 10.3934/dcdsb.2011.16.283
[18]	S. Yu. Pilyugin. Inverse shadowing by continuous methods. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 29-38. doi: 10.3934/dcds.2002.8.29
[19]	Scott G. McCalla. Paladins as predators: Invasive waves in a spatial evolutionary adversarial game. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1437-1457. doi: 10.3934/dcdsb.2014.19.1437
[20]	John Cleveland. Basic stage structure measure valued evolutionary game model. Mathematical Biosciences & Engineering, 2015, 12 (2) : 291-310. doi: 10.3934/mbe.2015.12.291

2017 Impact Factor: 1.219

American Institute of Mathematical Sciences