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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  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, 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

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