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

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

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  • 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.

    Mathematics Subject Classification: 35Q91, 35F20, 91A26.

    Citation:

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  • Figure 1.  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).

    Figure 2.  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 $.

    Figure 3.  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).

    Figure 4.  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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