There is much debate about how humans' decision-making compares with that of other primates. One way to explore this is to compare species' performance using identical methodologies in games with strategical interactions. We presented a computerized Assurance Game, which was either functionally simultaneous or sequential, to investigate how humans, rhesus monkeys and capuchin monkeys used information in decision-making. All species coordinated via sequential play on the payoff-dominant Nash equilibrium, indicating that information about the partner's choice improved decisions. Furthermore, some humans and rhesus monkeys found the payoff-dominant Nash equilibrium in the simultaneous game, even when it was the first condition presented. Thus, Old World primates solved the task without any external cues to their partner's choice. Finally, when not explicitly prohibited, humans spontaneously used language to coordinate on the payoff-dominant Nash equilibrium, indicating an alternative mechanism for converting a simultaneous move game into a sequential move game. This phylogenetic distribution implies that no single mechanism drives coordination decisions across the primates, while humans' ability to spontaneously use language to change the structure of the game emphasizes that multiple mechanisms may be used even within the same species. These results provide insight into the evolution of decision-making strategies across the primates.