A model of cultural evolution in the context of strategic conflict

Misha Perepelitsa

doi:10.3934/krm.2021014

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2021, Volume 14, Issue 3: 523-539. Doi: 10.3934/krm.2021014

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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: May 31, 2020

Revised: December 31, 2020

Early access: March 2021

Published: June 2021

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Abstract

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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Mathematics Subject Classification: 35L02, 35Q91, 91A22.

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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 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 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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