Game Dynamics as the Meaning of a Game

Learning dynamics have traditionally taken a secondary role to Nash equilibria in game theory. We propose a new approach that places the understanding of game dynamics over mixed strategy profiles as the central object of inquiry. We focus on the stable recurrent points of the dynamics, i.e. states which are likely to be revisited infinitely often; obviously, pure Nash equilibria are a special case of such behavior. We propose a new solution concept, the Markov-Conley Chain (MCC), which has several favorable properties: It is a simple randomized generalization of the pure Nash equilibrium, just like the mixed Nash equilibrium; every game has at least one MCC; an MCC is invariant under additive constants and positive multipliers of the players' utilities; there is a polynomial number of MCCs in any game, and they can be all computed in polynomial time; the MCCs can be shown to be, in a well defined sense, surrogates or traces of an important but elusive topological object called the sink chain component of the dynamics; finally, it can be shown that a natural game dynamics surely ends up at one of the MCCs of the game.