Probabilistic Interpretations of Integrability for Game Dynamics

In models of evolution and learning in games, a variety of proofs of convergence rely on the assumption that the players’ choice functions are integrable. This assumption does not have an obvious game-theoretic interpretation. We address this question by introducing probability models defined in terms of piecewise-smooth closed curves through $\mathbb{R}^{n}$ ; these curves describe cycles in the performances of the available actions. We establish that a choice function is integrable if and only if in the probability model induced by each such curve, the rate at which players switch to a randomly drawn action is uncorrelated with a certain binary signal. The binary signal specifies whether the performance of the randomly drawn action is improving or worsening, and can also be interpreted as a signal about the performances of actions other than the one randomly drawn. Copyright Springer Science+Business Media New York 2014