# American Institute of Mathematical Sciences

February  2021, 14(1): 115-148. doi: 10.3934/krm.2020051

## Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations

 IMAS (UBA-CONICET) and Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina

Received  March 2020 Revised  August 2020 Published  February 2021 Early access  November 2020

Fund Project: This work was partially supported by Universidad de Buenos Aires under grants 20020170100445BA and 20020170200256BA, by Agencia Nacional de Promocion Cientifica y Tecnica PICT 201-0215 and PICT 2016 1022

In this work we propose a kinetic formulation for evolutionary game theory for zero sum games when the agents use mixed strategies. We start with a simple adaptive rule, where after an encounter each agent increases by a small amount $h$ the probability of playing the successful pure strategy used in the match. We derive the Boltzmann equation which describes the macroscopic effects of this microscopical rule, and we obtain a first order, nonlocal, partial differential equation as the limit when $h$ goes to zero.

We study the relationship between this equation and the well known replicator equations, showing the equivalence between the concepts of Nash equilibria, stationary solutions of the partial differential equation, and the equilibria of the replicator equations. Finally, we relate the long-time behavior of solutions to the partial differential equation and the stability of the replicator equations.

Erratum: The month information has been corrected from January to February. We apologize for any inconvenience this may cause.

Citation: Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic and Related Models, 2021, 14 (1) : 115-148. doi: 10.3934/krm.2020051
##### References:
 [1] A. S. Ackleh and N. Saintier, Well-posedness of a system of transport and diffusion equations in space of measures, Journal of Mathematical Analysis and Applications, 492 (2020), 28 pp. doi: 10.1016/j.jmaa.2020.124397. [2] G. Albi, L. Pareschi and M. Zanella, Boltzmann games in heterogeneous consensus dynamics, Journal of Statistical Physics, 175 (2019), 97-125.  doi: 10.1007/s10955-019-02246-y. [3] R. Alonso, I. M. Gamba and M.-B. Tran, The Cauchy problem and BEC stability for the quantum Boltzmann-condensation system for bosons at very low temperature, arXiv preprint arxiv: 1609.07467, (2016). [4] L. Ambrosio, M. Fornasier, M. Morandotti and G. Savare, Spatially inhomogeneous evolutionary games, arXiv preprint arxiv: 1805.04027, (2018). [5] L. Arlotti, N. Bellomo and and E. De Angelis, Generalized kinetic (Boltzmann) models: Mathematical structures and applications, Mathematical Models and Methods in Applied Sciences, 12 (2002), 567-591.  doi: 10.1142/S0218202502001799. [6] N. Bellomo, Modeling Complex Living Systems: A Kinetic Theory and Stochastic Game Approach, Birkhäuser Boston, Inc., Boston, MA, 2008. [7] G. A. Marsan, N. Bellomo and A. Tosin, Complex Systems and Society: Modeling and Simulation, Springer, 2013. doi: 10.1007/978-1-4614-7242-1. [8] A. Boccabella, R. Natalini and L. Pareschi, On a continuous mixed strategies model for evolutionary game theory, Kinetic and Related Models, 4 (2011), 187-213.  doi: 10.3934/krm.2011.4.187. [9] W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the $1/n$ limit of interacting classical particles, Communications in Mathematical Physics, 56 (1977), 101-113.  doi: 10.1007/BF01611497. [10] A. Bressan, Notes on the Boltzmann Equation, Lecture notes for a summer course, SISSA Trieste, (2005). [11] J. A. Canizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Mathematical Models and Methods in Applied Sciences, 21 (2011), 515-539.  doi: 10.1142/S0218202511005131. [12] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, vol. 106, Springer Science & Business Media, 2013. [13] J. Cleveland and A. S. Ackleh, Evolutionary game theory on measure spaces: Well-posedness, Nonlinear Analysis: Real World Applications, 14 (2013), 785-797.  doi: 10.1016/j.nonrwa.2012.08.002. [14] 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. [15] ——, The Stability Concept of Evolutionary Game Theory: A Dynamic Approach, vol. 94, Springer Science & Business Media, 2013. [16] P. Degond and B. Lucquin-Desreux, The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case, Mathematical Models and Methods in Applied Sciences, 2 (1992), 167-182.  doi: 10.1142/S0218202592000119. [17] L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing, Transport Theory and Statistical Physics, 21 (1992), 259-276.  doi: 10.1080/00411459208203923. [18] L. Desvillettes and V. Ricci, A rigorous derivation of a linear kinetic equation of Fokker-Planck type in the limit of grazing collisions, Journal of Statistical Physics, 104 (2001), 1173-1189.  doi: 10.1023/A:1010461929872. [19] R. L. Dobrushin, Vlasov equations, Functional Analysis and Its Applications, 13 (1979), 48-58.  doi: 10.1007/BF01077243. [20] G. Gabetta, G. Toscani and B. Wennberg, Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation, Journal of statistical physics, 81 (1995), 901-934.  doi: 10.1007/BF02179298. [21] J. Geanakoplos, Nash and Walras equilibrium via Brouwer, Economic Theory, 21 (2003), 585-603.  doi: 10.1007/s001990000076. [22] F. Golse, On the dynamics of large particle systems in the mean field limit, in Macroscopic and large scale phenomena: Coarse graining, mean field limits and ergodicity, Springer, 2016, 1-144. doi: 10.1007/978-3-319-26883-5_1. [23] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.  doi: 10.1017/CBO9781139173179. [24] J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bulletin of the American Mathematical Society, 40 (2003), 479-519.  doi: 10.1090/S0273-0979-03-00988-1. [25] R. Laraki, J. Renault and S. Sorin, Mathematical Foundations of Game Theory, Springer, Cham, (2019). doi: 10.1007/978-3-030-26646-2. [26] G. A. Marsan, N. Bellomo and L. Gibelli, Stochastic evolutionary differential games toward a systems theory of behavioral social dynamics, Mathematical Models and Methods in Applied Sciences, 26 (2016), 1051-1093.  doi: 10.1142/S0218202516500251. [27] S. Mendoza-Palacios and O. Hernández-Lerma, Evolutionary dynamics on measurable strategy spaces: Asymmetric games, Journal of Differential Equations, 259 (2015), 5709-5733.  doi: 10.1016/j.jde.2015.07.005. [28] J. Miekisz, Evolutionary game theory and population dynamics, in Multiscale Problems in the Life Sciences, Springer, 2008, 269-316. doi: 10.1007/978-3-540-78362-6_5. [29] G. Naldi, L. Pareschi and G. Toscani, Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Springer Science & Business Media, 2010. doi: 10.1007/978-0-8176-4946-3. [30] H. Neunzert and J. Wick, Die approximation der Lösung von integro-differentialgleichungen durch endliche punktmengen, in Numerische Behandlung nichtlinearer Integrodifferential-und Differentialgleichungen, Springer, 1974, 275-290. [31] J. Oechssler and F. Riedel, Evolutionary dynamics on infinite strategy spaces, Economic Theory, 17 (2001), 141-162.  doi: 10.1007/PL00004092. [32] S. Onn and I. Weissman, Generating uniform random vectors over a simplex with implications to the volume of a certain polytope and to multivariate extremes., Ann Oper Res, 189 (2011), 331-342.  doi: 10.1007/s10479-009-0567-7. [33] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, OUP Oxford, 2013. [34] L. Pedraza, J. P. Pinasco and N. Saintier, Measure-valued opinion dynamics, Mathematical Models and Methods in Applied Sciences, 30 (2020), 225-260.  doi: 10.1142/S0218202520500062. [35] M. Pérez-Llanos, J. P. Pinasco, and N. Saintier, Opinion fitness and convergence to consensus in homogeneous and heterogeneous populations, 2020. [36] M. Pérez-Llanos, J. P. Pinasco, N. Saintier and A. Silva, Opinion formation models with heterogeneous persuasion and zealotry, SIAM Journal on Mathematical Analysis, 50 (2018), 4812-4837.  doi: 10.1137/17M1152784. [37] J. P. Pinasco, M. R. Cartabia and N. Saintier, A game theoretic model of wealth distribution, Dynamic Games and Applications, 8 (2018), 874-890.  doi: 10.1007/s13235-018-0240-3. [38] F. Salvarani and D. Tonon, Kinetic Description of Strategic Binary Games, 2019. [39] W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT press, 2010. [40] P. Schuster and K. Sigmund, Replicator dynamics, Journal of Theoretical Biology, 100 (1983), 533-538.  doi: 10.1016/0022-5193(83)90445-9. [41] F. Slanina and H. Lavicka, Analytical results for the Sznajd model of opinion formation, The European Physical Journal B, 35 (2003), 279-288.  doi: 10.1140/epjb/e2003-00278-0. [42] J. M. Smith, Evolution and the Theory of Games, Cambridge university press, 1982.  doi: 10.1017/CBO9780511806292. [43] P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics, Mathematical Biosciences, 40 (1978), 145-156.  doi: 10.1016/0025-5564(78)90077-9. [44] G. Toscani, Kinetic models of opinion formation, Communications in Mathematical Sciences, 4 (2006), 481-496.  doi: 10.4310/CMS.2006.v4.n3.a1. [45] A. Tosin, Kinetic equations and stochastic game theory for social systems, in Mathematical Models and Methods for Planet Earth, Springer, 2014, 37-57. doi: 10.1007/978-3-319-02657-2_4. [46] C. Villani, Topics in Optimal Transportation, American Mathematical Soc., 2003. doi: 10.1090/gsm/058. [47] A. Wornow, Generating random numebrs on a simplex, Computers and Geosciences, 19 (1993), 81-88.  doi: 10.1016/0098-3004(93)90045-7.

show all references

##### References:
 [1] A. S. Ackleh and N. Saintier, Well-posedness of a system of transport and diffusion equations in space of measures, Journal of Mathematical Analysis and Applications, 492 (2020), 28 pp. doi: 10.1016/j.jmaa.2020.124397. [2] G. Albi, L. Pareschi and M. Zanella, Boltzmann games in heterogeneous consensus dynamics, Journal of Statistical Physics, 175 (2019), 97-125.  doi: 10.1007/s10955-019-02246-y. [3] R. Alonso, I. M. Gamba and M.-B. Tran, The Cauchy problem and BEC stability for the quantum Boltzmann-condensation system for bosons at very low temperature, arXiv preprint arxiv: 1609.07467, (2016). [4] L. Ambrosio, M. Fornasier, M. Morandotti and G. Savare, Spatially inhomogeneous evolutionary games, arXiv preprint arxiv: 1805.04027, (2018). [5] L. Arlotti, N. Bellomo and and E. De Angelis, Generalized kinetic (Boltzmann) models: Mathematical structures and applications, Mathematical Models and Methods in Applied Sciences, 12 (2002), 567-591.  doi: 10.1142/S0218202502001799. [6] N. Bellomo, Modeling Complex Living Systems: A Kinetic Theory and Stochastic Game Approach, Birkhäuser Boston, Inc., Boston, MA, 2008. [7] G. A. Marsan, N. Bellomo and A. Tosin, Complex Systems and Society: Modeling and Simulation, Springer, 2013. doi: 10.1007/978-1-4614-7242-1. [8] A. Boccabella, R. Natalini and L. Pareschi, On a continuous mixed strategies model for evolutionary game theory, Kinetic and Related Models, 4 (2011), 187-213.  doi: 10.3934/krm.2011.4.187. [9] W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the $1/n$ limit of interacting classical particles, Communications in Mathematical Physics, 56 (1977), 101-113.  doi: 10.1007/BF01611497. [10] A. Bressan, Notes on the Boltzmann Equation, Lecture notes for a summer course, SISSA Trieste, (2005). [11] J. A. Canizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Mathematical Models and Methods in Applied Sciences, 21 (2011), 515-539.  doi: 10.1142/S0218202511005131. [12] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, vol. 106, Springer Science & Business Media, 2013. [13] J. Cleveland and A. S. Ackleh, Evolutionary game theory on measure spaces: Well-posedness, Nonlinear Analysis: Real World Applications, 14 (2013), 785-797.  doi: 10.1016/j.nonrwa.2012.08.002. [14] 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. [15] ——, The Stability Concept of Evolutionary Game Theory: A Dynamic Approach, vol. 94, Springer Science & Business Media, 2013. [16] P. Degond and B. Lucquin-Desreux, The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case, Mathematical Models and Methods in Applied Sciences, 2 (1992), 167-182.  doi: 10.1142/S0218202592000119. [17] L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing, Transport Theory and Statistical Physics, 21 (1992), 259-276.  doi: 10.1080/00411459208203923. [18] L. Desvillettes and V. Ricci, A rigorous derivation of a linear kinetic equation of Fokker-Planck type in the limit of grazing collisions, Journal of Statistical Physics, 104 (2001), 1173-1189.  doi: 10.1023/A:1010461929872. [19] R. L. Dobrushin, Vlasov equations, Functional Analysis and Its Applications, 13 (1979), 48-58.  doi: 10.1007/BF01077243. [20] G. Gabetta, G. Toscani and B. Wennberg, Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation, Journal of statistical physics, 81 (1995), 901-934.  doi: 10.1007/BF02179298. [21] J. Geanakoplos, Nash and Walras equilibrium via Brouwer, Economic Theory, 21 (2003), 585-603.  doi: 10.1007/s001990000076. [22] F. Golse, On the dynamics of large particle systems in the mean field limit, in Macroscopic and large scale phenomena: Coarse graining, mean field limits and ergodicity, Springer, 2016, 1-144. doi: 10.1007/978-3-319-26883-5_1. [23] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.  doi: 10.1017/CBO9781139173179. [24] J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bulletin of the American Mathematical Society, 40 (2003), 479-519.  doi: 10.1090/S0273-0979-03-00988-1. [25] R. Laraki, J. Renault and S. Sorin, Mathematical Foundations of Game Theory, Springer, Cham, (2019). doi: 10.1007/978-3-030-26646-2. [26] G. A. Marsan, N. Bellomo and L. Gibelli, Stochastic evolutionary differential games toward a systems theory of behavioral social dynamics, Mathematical Models and Methods in Applied Sciences, 26 (2016), 1051-1093.  doi: 10.1142/S0218202516500251. [27] S. Mendoza-Palacios and O. Hernández-Lerma, Evolutionary dynamics on measurable strategy spaces: Asymmetric games, Journal of Differential Equations, 259 (2015), 5709-5733.  doi: 10.1016/j.jde.2015.07.005. [28] J. Miekisz, Evolutionary game theory and population dynamics, in Multiscale Problems in the Life Sciences, Springer, 2008, 269-316. doi: 10.1007/978-3-540-78362-6_5. [29] G. Naldi, L. Pareschi and G. Toscani, Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Springer Science & Business Media, 2010. doi: 10.1007/978-0-8176-4946-3. [30] H. Neunzert and J. Wick, Die approximation der Lösung von integro-differentialgleichungen durch endliche punktmengen, in Numerische Behandlung nichtlinearer Integrodifferential-und Differentialgleichungen, Springer, 1974, 275-290. [31] J. Oechssler and F. Riedel, Evolutionary dynamics on infinite strategy spaces, Economic Theory, 17 (2001), 141-162.  doi: 10.1007/PL00004092. [32] S. Onn and I. Weissman, Generating uniform random vectors over a simplex with implications to the volume of a certain polytope and to multivariate extremes., Ann Oper Res, 189 (2011), 331-342.  doi: 10.1007/s10479-009-0567-7. [33] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, OUP Oxford, 2013. [34] L. Pedraza, J. P. Pinasco and N. Saintier, Measure-valued opinion dynamics, Mathematical Models and Methods in Applied Sciences, 30 (2020), 225-260.  doi: 10.1142/S0218202520500062. [35] M. Pérez-Llanos, J. P. Pinasco, and N. Saintier, Opinion fitness and convergence to consensus in homogeneous and heterogeneous populations, 2020. [36] M. Pérez-Llanos, J. P. Pinasco, N. Saintier and A. Silva, Opinion formation models with heterogeneous persuasion and zealotry, SIAM Journal on Mathematical Analysis, 50 (2018), 4812-4837.  doi: 10.1137/17M1152784. [37] J. P. Pinasco, M. R. Cartabia and N. Saintier, A game theoretic model of wealth distribution, Dynamic Games and Applications, 8 (2018), 874-890.  doi: 10.1007/s13235-018-0240-3. [38] F. Salvarani and D. Tonon, Kinetic Description of Strategic Binary Games, 2019. [39] W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT press, 2010. [40] P. Schuster and K. Sigmund, Replicator dynamics, Journal of Theoretical Biology, 100 (1983), 533-538.  doi: 10.1016/0022-5193(83)90445-9. [41] F. Slanina and H. Lavicka, Analytical results for the Sznajd model of opinion formation, The European Physical Journal B, 35 (2003), 279-288.  doi: 10.1140/epjb/e2003-00278-0. [42] J. M. Smith, Evolution and the Theory of Games, Cambridge university press, 1982.  doi: 10.1017/CBO9780511806292. [43] P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics, Mathematical Biosciences, 40 (1978), 145-156.  doi: 10.1016/0025-5564(78)90077-9. [44] G. Toscani, Kinetic models of opinion formation, Communications in Mathematical Sciences, 4 (2006), 481-496.  doi: 10.4310/CMS.2006.v4.n3.a1. [45] A. Tosin, Kinetic equations and stochastic game theory for social systems, in Mathematical Models and Methods for Planet Earth, Springer, 2014, 37-57. doi: 10.1007/978-3-319-02657-2_4. [46] C. Villani, Topics in Optimal Transportation, American Mathematical Soc., 2003. doi: 10.1090/gsm/058. [47] A. Wornow, Generating random numebrs on a simplex, Computers and Geosciences, 19 (1993), 81-88.  doi: 10.1016/0098-3004(93)90045-7.
Time evolution of the distribution of $p^1_1, \ldots, p^N_1$ obtained solving the transport equation (5.12) (left) and from an agent-based simulation (right).
Time evolution of the distribution of $p^1_1, \ldots, p^N_1$, $N=1000$, in the agent-based simulation starting from a uniform distribution in $[0, 0.3]$ with $\delta=0.01$ and different values of noise level $r$.
Averaged proportion of agents in the agent-based simulation with $p\in [0, 0.01]$ at time $\tau=800$ in function of the noise level $r$ (the average is computed over 10 simulations).
Plot of $\log(\max\, p_1-\min\, p_1)$ in the agent-based simulation with $N=1000$, $\delta=c=0.01$, $r=0$ and two different functions $h$.
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