We introduce a two-player, binary-choice game in which both players have a privately known incentive to enter, yet the combined surplus is highest if only one enters. Repetition of this game admits two distinct ways to cooperate: turn taking and cutoffs, which rely on the player's private value to entry. A series of experiments highlights the role of private information in determining which mode players adopt. If an individual's entry values vary little (e.g. mundane tasks), taking turns is likely; if these potential values are diverse (e.g. difficult tasks that differentiate individuals by skill or preferences), cutoff cooperation emerges.