As an agent gathers information about its environment and monitors the decisions of other agents, its behavior may fluctuate adaptively over short time scales while still maintaining a long-term strategy. We designed a real-time virtual environment to experimentally investigate the relationship between the micro-level dynamics of dyadic behavior within single games and the macro-level dynamics of outcomes across iterated games. In one experiment, participants played a real-time game of "chicken," simultaneously guiding avatars toward high-payoff or low-payoff targets. If both participants reached a demarcated vicinity of a target at the same time, that target was destroyed. We recorded their trajectories, and induced uncertainty by adding noise to their movement speeds. At the macro-level, we found evidence of self-organized turn-taking across repeated games. At the micro-level, we found that even within a turn-taking equilibrium, both players competitively pursued the high payoff for a period of time before one of them diverted.