Does learning converge to equilibrium in generic games?

Game theory models interacting biological and social systems. In a repeated game, players may converge or their dynamics oscillate. If the system is not designed to converge, which of these two behaviors can we expect? I will give an overview of research that addresses this question by studying convergence to equilibrium in generic games. I will consider simple 2-player, 2-action games, where players learn through Experience-Weighted Attraction learning. We are able to characterize convergence in the space of parameters and games. Games with a “best-reply cycle”, such as Matching Pennies, are the ones in which convergence is less likely. Further, I will show that the frequency of best-reply cycles predicts convergence of six learning rules in 2-player games with an arbitrary number of actions, when games are sampled at random given constraints.The larger the number of actions, or the more anti-correlated the payoffs, the more best-reply cycles become dominant, and convergence becomes less likely. Finally, I consider games with an arbitrary number of players and with a network structure. I show that more players and more dense networks increase the importance of best-reply cycles, making convergence unlikely. Overall, these results indicate that in many cases equilibrium is an unrealistic assumption, and one must explicitly model the dynamics of learning.

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Tobias Galla

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