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A mean-field limit of the particle swarmalator model
A model of cultural evolution in the context of strategic conflict
Department of Mathematics, University of Houston, 4800 Calhoun Rd, Houston, TX |
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.
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A note on evolutionary stable strategies and game dynamics, J. Theoret. Biol., 81 (1979), 609-612.
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J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.
doi: 10.1017/CBO9781139173179.![]() ![]() |
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doi: 10.23943/princeton/9780691150703.001.0001.![]() |
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J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, New York, 1982.
doi: 10.1017/CBO9780511806292.![]() |
[12] |
J. Maynard Smith and G. R. Price,
The logic of animal conflict, Nature, 246 (1973), 15-18.
doi: 10.1038/246015a0. |
[13] |
M. Perepelitsa,
Adaptive learning in large populations, J. Math. Biol., 79 (2019), 2237-2253.
doi: 10.1007/s00285-019-01427-3. |
[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.![]() |
[15] |
J. Robinson,
An iterative method of solving a game, Ann. of Math. (2), 54 (1951), 296-301.
doi: 10.2307/1969530. |
[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. |
[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. |
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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. |
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. |
[2] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
doi: 10.1007/978-0-387-70914-7. |
[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. |
[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. |
[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. |
[6] |
I. Gilboa and A. Matsui,
Social stability and equilibrium, Econometrica, 59 (1991), 859-867.
doi: 10.2307/2938230. |
[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. |
[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. |
[9] |
J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.
doi: 10.1017/CBO9781139173179.![]() ![]() |
[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.![]() |
[11] |
J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, New York, 1982.
doi: 10.1017/CBO9780511806292.![]() |
[12] |
J. Maynard Smith and G. R. Price,
The logic of animal conflict, Nature, 246 (1973), 15-18.
doi: 10.1038/246015a0. |
[13] |
M. Perepelitsa,
Adaptive learning in large populations, J. Math. Biol., 79 (2019), 2237-2253.
doi: 10.1007/s00285-019-01427-3. |
[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.![]() |
[15] |
J. Robinson,
An iterative method of solving a game, Ann. of Math. (2), 54 (1951), 296-301.
doi: 10.2307/1969530. |
[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. |
[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. |
[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. |




Hawk | Dove | Retaliator | |
Hawk | -1 | 2 | -1 |
Dove | 0 | 1 | 0.9 |
Retaliator | -1 | 1.1 | 1 |
Hawk | Dove | Retaliator | |
Hawk | -1 | 2 | -1 |
Dove | 0 | 1 | 0.9 |
Retaliator | -1 | 1.1 | 1 |
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