Pure Nash Equilibria and Best-Response Dynamics in Random Games

05/26/2019 ∙ by Ben Amiet, et al. ∙ 0

Nash equilibria are a central concept in game theory and have applications in fields such as economics, evolutionary biology, theoretical computer science, and many others. Mixed equilibria exist in any finite game, but pure equilibria may fail to exist. We consider the existence of pure Nash equilibria in games where the payoffs are drawn at random. In particular, we consider games where a large number of players can each choose one of two possible actions, and the payoffs are i.i.d. with the possibility of ties. We provide asymptotic results about the random number of pure Nash equilibria, such as fast growth and a central limit theorem. Moreover, we establish a new link between percolation models and game theory to shed light on various aspects of Nash equilibria. Through this connection, we describe in detail the geometry of Nash equilibria and show that, when the probability of ties is small, a best-response dynamics reaches a Nash equilibrium with a probability that quickly approaches one as the number of players grows. We show a multitude of phase transitions depending on a single parameter of the model, that is, the probability of having ties.

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