In common interest games, players generally manage to coordinate their actions on mutually optimal outcomes, but orthodox game theory provides no reason for them to play their individual parts in these seemingly obvious solutions and no justification for choosing the corresponding strategies. A number of theories have been suggested to explain coordination, among the most prominent being versions of cognitive hierarchy theory, theories of team reasoning, and social projection theory (in symmetric games). Each of these theories provides a plausible explanation but is theoretically problematic. An improved theory of strong Stackelberg reasoning avoids these problems and explains coordination among players who care about their co-players’ payoffs and who act as though their co-players can anticipate their choices. Two experiments designed to test cognitive hierarchy, team reasoning, and strong Stackelberg theories against one another in games without obvious, payoff-dominant solutions suggest that each of the theories provides part of the explanation. Cognitive hierarchy Level-1 reasoning, facilitated by a heuristic of avoiding the worst payoff, tended to predominate, especially in more complicated games, but strong Stackelberg reasoning occurred quite frequently in the simpler games and team reasoning in both the simpler and the more complicated games. Most players considered 2 or more of these reasoning processes before choosing their strategies. (PsycINFO Database Record (c) 2016 APA, all rights reserved)