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Survival of dominated strategies under imitation dynamics

  • *Corresponding author: Yannick Viossat

    *Corresponding author: Yannick Viossat

The first author is grateful for financial support by the French National Research Agency (ANR) in the framework of the "Investissements d'avenir" program (ANR-15-IDEX-02), the LabEx PERSYVAL (ANR-11-LABX-0025-01), MIAI@Grenoble Alpes (ANR-19-P3IA-0003), and the bilateral ANR-NRF grant ALIAS (ANR-19-CE48-0018-01)

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  • The literature on evolutionary game theory suggests that pure strategies that are strictly dominated by other pure strategies always become extinct under imitative game dynamics, but they can survive under innovative dynamics. As we explain, this is because innovative dynamics favour rare strategies while standard imitative dynamics do not. However, as we also show, there are reasonable imitation protocols that favour rare or frequent strategies, thus allowing strictly dominated strategies to survive in large classes of imitation dynamics. Dominated strategies can persist at nontrivial frequencies even when the level of domination is not small.

    Mathematics Subject Classification: Primary: 91A22, 91A26.

    Citation:

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  • Figure 1.  Asymptotic frequency of the dominated strategy as a function of the payoff ratio $ u_2/u_1 $ for various values of $ m $

    Figure 2.  Imitation dynamics in Game (6). Top panel: $ d = 0.04 $. Bottom panel: $ d = 0.08 $. In blue and green, two solutions with respective initial conditions $ (1/7, 2/7, 1/7, 3/7) $ and $ (1/7, 1/7, 4/7, 1/7) $. In red (hardly visible on the top panel), what appears to be a common limit cycle. The dynamics are described in the main text

    Figure 3.  Imitation dynamics favouring rare strategies in Game (9) against an oscillating behavior of the opponent. Top-panels: the solution $ x(t) $. The point $ P_i $ corresponds to the population state where everybody plays strategy $ i $. Bottom-panels: frequency of the strictly dominated strategy (strategy 3). Left-column: $ \varepsilon = 0.05 $. Right-column: $ \varepsilon = 0.1 $. In blue and green, two solutions with respective initial conditions $ (1/3, 1/6, 1/2) $ and $ (1/6, 2/3, 1/6) $

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