Logit-response dynamics (Alos-Ferrer and Netzer, Games and Economic Behavior 2010) are a rich and natural class of noisy best-response dynamics. In this work we revise the price of anarchy and the price of stability by considering the quality of long-run equilibria in these dynamics. Our results show that prior studies on simpler dynamics of this type can strongly depend on a synchronous schedule of the players' moves. In particular, a small noise by itself is not enough to improve the quality of equilibria as soon as other very natural schedules are used.
We develop a characterization of stochastically stable states for the logit-response learning dynamics in games, with arbitrary specification of revision opportunities. The result allows us to show convergence to the set of Nash equilibria in the class of best-response potential games and the failure of the dynamics to select potential maximizers beyond the class of exact potential games. We also study to which extent equilibrium selection is robust to the specification of revision opportunities. Our techniques can be extended and applied to a wide class of learning dynamics in games.