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Cyclical behavior of evolutionary dynamics in coordination games with changing payoffs


The paper presents a model of two-speed evolution in which the payoffs in the population game (or, alternatively, the individual preferences) slowly adjust to changes in the aggregate behavior of the population. The model investigates how, for a population of myopic agents with homogeneous preferences, changes in the environment caused by current aggregate behavior may affect future payoffs and hence alter future behavior. The interaction between the agents is based on a symmetric two-strategy game with positive externalities and negative feedback from aggregate behavior to payoffs, so that at every point in time the population has an incentive to coordinate, whereas over time the more popular strategy becomes less appealing. Under the best response dynamics and the logit dynamics with small noise levels the joint trajectories of preferences and behavior converge to closed orbits around the unique steady state, whereas for large noise levels the steady state of the logit dynamics becomes a sink. Under the replicator dynamics the unique steady state of the system is repelling and the trajectories are unbounded unstable spirals.

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  1. See Sandholm (2010) for background on evolutionary games and Newton (2018) for the survey on the current state of the field.

  2. A number of models introducing environmental feedback into evolutionary games has been developed in biology. See, for instance, Akçay and Roughgarden (2011), Weitz et al. (2016), and Tilman et al. (2020).

  3. See Vanderbilt (2009) which documents the steps Facebook had to take in order to keep up with the constantly increasing number of users and amount of user-produced data.

  4. In coordination games this state coincides with the mixed strategy equilibrium. In games with a dominant strategy this point lies outside the unit interval.

  5. While our interest is motivated by cases in which preferences evolve slower than behavior (r is close to 0), our results are qualitatively the same for all positive values of r.

  6. See Björnerstedt and Weibull (1996) and Schlag (1998).

  7. I thank Matthew Johnston for discovering it.


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I thank Antonio Penta, Dan Quint, Keith Paarporn, and two anonymous referees for their comments and suggestions; Matthew Johnston and Vasily Zemchikhin for helping me with Lyapunov analysis. I am especially grateful to my advison Bill Sandholm (1970–2020) for his time, support, and encouragement during all stages of this project.

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Correspondence to George Loginov.

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Loginov, G. Cyclical behavior of evolutionary dynamics in coordination games with changing payoffs. Int J Game Theory 51, 1–27 (2022).

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  • Evolutionary game dynamics
  • Environmental feedback
  • Two speed evolution
  • Evolution of preferences