This paper provides deterministic approximation results for stochastic processes that arise when finite populations of boundedly rational agents recurrently play finite games. The deterministic approximation is defined in continuous time in terms of a system of ordinary differential equations of the type studied in evolutionary game theory. We establish precise connections between the long-run behavior of the stochastic process, for large populations, and its deterministic approximation. In particular, we show that if the deterministic flow enters a basin of attraction, then the stochastic process follows this flow closely until this (deterministic) entry time, with a probability that approaches one exponentially in the population size. After entry, the process remains in a neighborhood of the attractor for a random time span that exceeds an exponential function of the population size. The process spends almost all this time in a neighborhood of a subset of the attractor, the Birkhoff center of the flow restricted to the attractor.