American Institute of Mathematical Sciences

Advanced
June  2021, 14(3): 523-539. doi: 10.3934/krm.2021014

A model of cultural evolution in the context of strategic conflict

Misha Perepelitsa

Department of Mathematics, University of Houston, 4800 Calhoun Rd, Houston, TX

Received  June 2020 Revised  January 2021 Published  June 2021 Early access  March 2021

We consider a model of cultural evolution for a strategy selection in a population of individuals who interact in a game theoretic framework. The evolution combines individual learning of the environment (population strategy profile), reproduction, proportional to the success of the acquired knowledge, and social transmission of the knowledge to the next generation. A mean-field type equation is derived that describes the dynamics of the distribution of cultural traits, in terms of the rate of learning, the reproduction rate and population size. We establish global well-posedness of the initial-boundary value problem for this equation and give several examples that illustrate the process of the cultural evolution.

Keywords: Kinetic models in population dynamics, Fokker-Planck equation, multi-agent game theory, cultural evolution, learning in games.
Mathematics Subject Classification: 35L02, 35Q91, 91A22.
Citation: Misha Perepelitsa. A model of cultural evolution in the context of strategic conflict. Kinetic & Related Models, 2021, 14 (3) : 523-539. doi: 10.3934/krm.2021014
References:
[1]

T. Börgers and R. Sarin, Learning through reinforcement and replicator dynamics, J. Econom. Theory, 77 (1997), 1-14.  doi: 10.1006/jeth.1997.2319.  Google Scholar

[2]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. doi: 10.1007/978-0-387-70914-7.  Google Scholar

[3]

K. Chatterjee, D. Zufferey and M. A. Nowak, Evolutionary game dynamics in populations with different learners, J. Theoret. Biol., 301 (2012), 161-173.  doi: 10.1016/j.jtbi.2012.02.021.  Google Scholar

[4]

D. Fudenberg and D. K. Levine, The Theory of Learning in Games, MIT Press Series on Economic Learning and Social Evolution, 2, MIT Press, Cambridge, MA, 1998.  Google Scholar

[5]

A. Gaunersdorfer and J. Hofbauer, Fictitious play, Shapley polygons and the replicator equation. Evolutionary game theory in biology and economics, Games Econom. Behav., 11 (1995), 279-303.  doi: 10.1006/game.1995.1052.  Google Scholar

[6]

I. Gilboa and A. Matsui, Social stability and equilibrium, Econometrica, 59 (1991), 859-867.  doi: 10.2307/2938230.  Google Scholar

[7]

J. Hofbauer, From Nash and Brown to Maynard Smith: Equilibria, dynamics and ESS, Selection, 1 (2000), 81-88.  doi: 10.1556/Select.1.2000.1-3.8.  Google Scholar

[8]

J. Hofbauer, P. Schuster and K. Sigmund, A note on evolutionary stable strategies and game dynamics, J. Theoret. Biol., 81 (1979), 609-612.  doi: 10.1016/0022-5193(79)90058-4.  Google Scholar

[9] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9781139173179.  Google Scholar
[10] W. Hoppitt and K. Laland, Social Learning: An Introduction to Mechanisms, Methods and Models, Princenton University Press, 2013.  doi: 10.23943/princeton/9780691150703.001.0001.  Google Scholar
[11] J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, New York, 1982.  doi: 10.1017/CBO9780511806292.  Google Scholar
[12]

J. Maynard Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18.  doi: 10.1038/246015a0.  Google Scholar

[13]

M. Perepelitsa, Adaptive learning in large populations, J. Math. Biol., 79 (2019), 2237-2253.  doi: 10.1007/s00285-019-01427-3.  Google Scholar

[14] P. Richerson and R. Boyd, Not by Genes Alone: How Culture Transformed Human Evolution, University of Chicago Press, 2004.  doi: 10.7208/chicago/9780226712130.001.0001.  Google Scholar
[15]

J. Robinson, An iterative method of solving a game, Ann. of Math. (2), 54 (1951), 296-301.  doi: 10.2307/1969530.  Google Scholar

[16]

L. S. Shapley, Some topics in two person games, in Advances in Game Theory, Princeton Univ. Press, Princeton, NJ, 1964, 1-28. doi: 10.1515/9781400882014-002.  Google Scholar

[17]

P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics, Math. Biosci., 40 (1978), 145-156.  doi: 10.1016/0025-5564(78)90077-9.  Google Scholar

[18]

E. C. Zeeman, Dynamics of the evolution of animal conflicts, J. Theoret. Biol., 89 (1981), 249-270.  doi: 10.1016/0022-5193(81)90311-8.  Google Scholar

show all references

References:
[1]

T. Börgers and R. Sarin, Learning through reinforcement and replicator dynamics, J. Econom. Theory, 77 (1997), 1-14.  doi: 10.1006/jeth.1997.2319.  Google Scholar

[2]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. doi: 10.1007/978-0-387-70914-7.  Google Scholar

[3]

K. Chatterjee, D. Zufferey and M. A. Nowak, Evolutionary game dynamics in populations with different learners, J. Theoret. Biol., 301 (2012), 161-173.  doi: 10.1016/j.jtbi.2012.02.021.  Google Scholar

[4]

D. Fudenberg and D. K. Levine, The Theory of Learning in Games, MIT Press Series on Economic Learning and Social Evolution, 2, MIT Press, Cambridge, MA, 1998.  Google Scholar

[5]

A. Gaunersdorfer and J. Hofbauer, Fictitious play, Shapley polygons and the replicator equation. Evolutionary game theory in biology and economics, Games Econom. Behav., 11 (1995), 279-303.  doi: 10.1006/game.1995.1052.  Google Scholar

[6]

I. Gilboa and A. Matsui, Social stability and equilibrium, Econometrica, 59 (1991), 859-867.  doi: 10.2307/2938230.  Google Scholar

[7]

J. Hofbauer, From Nash and Brown to Maynard Smith: Equilibria, dynamics and ESS, Selection, 1 (2000), 81-88.  doi: 10.1556/Select.1.2000.1-3.8.  Google Scholar

[8]

J. Hofbauer, P. Schuster and K. Sigmund, A note on evolutionary stable strategies and game dynamics, J. Theoret. Biol., 81 (1979), 609-612.  doi: 10.1016/0022-5193(79)90058-4.  Google Scholar

[9] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9781139173179.  Google Scholar
[10] W. Hoppitt and K. Laland, Social Learning: An Introduction to Mechanisms, Methods and Models, Princenton University Press, 2013.  doi: 10.23943/princeton/9780691150703.001.0001.  Google Scholar
[11] J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, New York, 1982.  doi: 10.1017/CBO9780511806292.  Google Scholar
[12]

J. Maynard Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18.  doi: 10.1038/246015a0.  Google Scholar

[13]

M. Perepelitsa, Adaptive learning in large populations, J. Math. Biol., 79 (2019), 2237-2253.  doi: 10.1007/s00285-019-01427-3.  Google Scholar

[14] P. Richerson and R. Boyd, Not by Genes Alone: How Culture Transformed Human Evolution, University of Chicago Press, 2004.  doi: 10.7208/chicago/9780226712130.001.0001.  Google Scholar
[15]

J. Robinson, An iterative method of solving a game, Ann. of Math. (2), 54 (1951), 296-301.  doi: 10.2307/1969530.  Google Scholar

[16]

L. S. Shapley, Some topics in two person games, in Advances in Game Theory, Princeton Univ. Press, Princeton, NJ, 1964, 1-28. doi: 10.1515/9781400882014-002.  Google Scholar

[17]

P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics, Math. Biosci., 40 (1978), 145-156.  doi: 10.1016/0025-5564(78)90077-9.  Google Scholar

[18]

E. C. Zeeman, Dynamics of the evolution of animal conflicts, J. Theoret. Biol., 89 (1981), 249-270.  doi: 10.1016/0022-5193(81)90311-8.  Google Scholar

table 1. On the top plot, three polygonal regions, formed by lines $ OM, $ $ ON $ and $ O(0.5H+0.5D) $ are the regions where the best response function a single value: $ H, $ $ D, $ or $ R, $ respectively. Line $ KON $ separates the basins of attraction of states $ R $ and $ 0.5H+0.5D. $ On the bottom plot line $ HOD $ separates the basins of attraction of the same states $ R $ and $ 0.5H+0.5D. $ The plots show two trajectories for the best-response and the replication equations that converge to $ R $ and $ 0.5H+0.5D, $ respectively">Figure 1.  Phase portraites for the best-response (top) and the replicator (bottom) equations for Hawk-Dove-Retaliator game in table 1. On the top plot, three polygonal regions, formed by lines $ OM, $ $ ON $ and $ O(0.5H+0.5D) $ are the regions where the best response function a single value: $ H, $ $ D, $ or $ R, $ respectively. Line $ KON $ separates the basins of attraction of states $ R $ and $ 0.5H+0.5D. $ On the bottom plot line $ HOD $ separates the basins of attraction of the same states $ R $ and $ 0.5H+0.5D. $ The plots show two trajectories for the best-response and the replication equations that converge to $ R $ and $ 0.5H+0.5D, $ respectively
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Cultural evolution: small learning-to-reproduction ratio $ \alpha_p/\alpha. $ Three sub-populations starting at $ A, $ $ B, $ and $ C, $ respectively, move toward Retaliator strategy in empirical frequencies $ p_i(t). $ Blue line is the trajectory of the mean best response (population strategy profile) $ \bar{b}(t). $ It changes discontinuously when one of the sub-populations moves into an adjacent decision polygon. When all sub-groups move into the upper polygon, $ \bar{b}(t) = R. $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. All groups sharply change their dynamics and converge to $ 0.5H+0.5D $ after the group that started at $ C $ moves to the adjacent polygon. Blue line is a discontinuous trajectory of the mean best response $ \bar{b}(t) $ (strategy profile). It starts at point $ Q, $ evolves continuously and then jumps to point $ Q1 $ and then proceed continuously to $ 0.5H+0.5D. $">Figure 3.  Cultural evolution: large learning-to-reproduction ratio $ \alpha_p/\alpha. $ Same initial conditions as in Figure 2. All groups sharply change their dynamics and converge to $ 0.5H+0.5D $ after the group that started at $ C $ moves to the adjacent polygon. Blue line is a discontinuous trajectory of the mean best response $ \bar{b}(t) $ (strategy profile). It starts at point $ Q, $ evolves continuously and then jumps to point $ Q1 $ and then proceed continuously to $ 0.5H+0.5D. $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. Sub-populations started at $ A, $ $ B $ and $ C $ do not leave their decision polygons. Trajectory starting at $ Q $ is the mean best-response (strategy profile) determined by the replication equations. Asymptotically it moves to $ R, $ meaning that the sub-population that started at point $ C $ out-evolves other sub-populations">Figure 4.  Cultural evolution with constant interaction frequency. Same initial conditions as in Figure 2. Sub-populations started at $ A, $ $ B $ and $ C $ do not leave their decision polygons. Trajectory starting at $ Q $ is the mean best-response (strategy profile) determined by the replication equations. Asymptotically it moves to $ R, $ meaning that the sub-population that started at point $ C $ out-evolves other sub-populations
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Hawk-Dove-Retaliator game
Hawk Dove Retaliator
Hawk -1 2 -1
Dove 0 1 0.9
Retaliator -1 1.1 1
Table Options

Download as excel

Hawk Dove Retaliator
Hawk -1 2 -1
Dove 0 1 0.9
Retaliator -1 1.1 1
[1]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401
[2]

Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic & Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056
[3]

Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250
[4]

John W. Barrett, Endre Süli. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 371-408. doi: 10.3934/dcdss.2010.3.371
[5]

Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic & Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165
[6]

Patrick Cattiaux, Elissar Nasreddine, Marjolaine Puel. Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case. Kinetic & Related Models, 2019, 12 (4) : 727-748. doi: 10.3934/krm.2019028
[7]

Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017
[8]

Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic & Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016
[9]

Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic & Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485
[10]

Yuan Gao, Guangzhen Jin, Jian-Guo Liu. Inbetweening auto-animation via Fokker-Planck dynamics and thresholding. Inverse Problems & Imaging, 2021, 15 (5) : 843-864. doi: 10.3934/ipi.2021016
[11]

Florian Schneider, Andreas Roth, Jochen Kall. First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions. Kinetic & Related Models, 2017, 10 (4) : 1127-1161. doi: 10.3934/krm.2017044
[12]

Seung-Yeal Ha, Dohyun Kim, Jaeseung Lee, Se Eun Noh. Emergent dynamics of an orientation flocking model for multi-agent system. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2037-2060. doi: 10.3934/dcds.2020105
[13]

Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008
[14]

Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks & Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028
[15]

Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028
[16]

Richard Carney, Monique Chyba, Chris Gray, George Wilkens, Corey Shanbrom. Multi-agent systems for quadcopters. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021005
[17]

Tyrone E. Duncan. Some partially observed multi-agent linear exponential quadratic stochastic differential games. Evolution Equations & Control Theory, 2018, 7 (4) : 587-597. doi: 10.3934/eect.2018028
[18]

Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215
[19]

Martin Burger, Ina Humpert, Jan-Frederik Pietschmann. On Fokker-Planck equations with In- and Outflow of Mass. Kinetic & Related Models, 2020, 13 (2) : 249-277. doi: 10.3934/krm.2020009
[20]

Giulia Cavagnari, Antonio Marigonda, Benedetto Piccoli. Optimal synchronization problem for a multi-agent system. Networks & Heterogeneous Media, 2017, 12 (2) : 277-295. doi: 10.3934/nhm.2017012

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (116)
  • HTML views (171)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy