Many human interactions involve patterns of turn-taking cooperation that can be modeled by the deeply paradoxical Centipede game. A backward induction argument suggests that cooperation is irrational in such interactions, but experiments have demonstrated that players cooperate frequently and earn better payoffs as a consequence. We formulate 6 competing theories of cooperation in Centipede games and report the results of 2 experiments, based on investigations of several closely matched games with different payoff structures and different methods of reaching decisions. The results show that turn-taking cooperation does not appear to be explained by reciprocity theory, activity bias theory, or a motive to maximize relative payoffs, but that collective rationality, in the form of a motive to maximize joint payoffs, and fuzzy-trace theory can explain cooperation in interactions of this type. Reciprocity increases cooperation across repeated games between fixed player pairs, but there is no evidence of reciprocity influencing cooperation within games.