Survival of dominated strategies under imitation dynamics

Panayotis Mertikopoulos; Yannick Viossat

doi:10.3934/jdg.2022021

Article Contents

2022, Volume 9, Issue 4: 499-528. Doi: 10.3934/jdg.2022021

This issue Previous Article "Test two, choose the better" leads to high cooperation in the Centipede game Next Article Stochastic dynamics and Edmonds' algorithm

Survival of dominated strategies under imitation dynamics

Panayotis Mertikopoulos^1, and
Yannick Viossat^2, ,

1.
Univ. Grenoble Alpes, CNRS, Inria, LIG, 38000, Grenoble, France
2.
CEREMADE, Université Paris Dauphine-PSL, Place du Maréchal de Lattre de Tassigny, F-75775 Paris, France

^*Corresponding author: Yannick Viossat
^*Corresponding author: Yannick Viossat

Received: November 2021

Revised: August 2022

Early access: September 2022

Published: October 2022

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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Abstract

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.

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

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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 $

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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

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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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