John F Nash (1950) proposed dynamics for repeated interactions according to which agents myopically play individual best-responses against their observations of other agents’ past play. Such dynamics converge to Nash equilibria. Without suitable mechanisms, this means that best-response dynamics can lead to low levels of cooperative behavior and thus to inefficient outcomes in social dilemma games. Here, we discuss the theoretical predictions of these dynamics in a variety of social dilemmas and assess these in light of behavioral evidence. We particularly focus on “meritocratic matching”, a class of mechanisms that leads to both low cooperation (inefficient) and high cooperation (near-efficient) equilibria (Gunnthorsdottir et al. 2010; Nax, Murphy, and Helbing 2014; Nax et al. 2015). Most behavioral theories derived from related social dilemmas cannot explain the behavioral evidence for this class of games, but Nash dynamics provide a very satisfactory explanation. We also argue that Nash dynamics provide a parsimonious account of behavioral results for several different social dilemmas, with the exception of the linear public goods game.