In common interest games in which players are motivated to coordinate their strategies to achieve a jointly optimal outcome, orthodox game theory provides no general reason or justification for choosing the required strategies. In the simplest cases, where the optimal strategies are intuitively obvious, human decision makers generally coordinate without difficulty, but how they achieve this is poorly understood. Most theories seeking to explain strategic coordination have limited applicability, or require changes to the game specification, or introduce implausible assumptions or radical departures from fundamental game-theoretic assumptions. The theory of strong Stackelberg reasoning, according to which players choose strategies that would maximize their own payoffs if their co-players could invariably anticipate any strategy and respond with a best reply to it, avoids these problems and explains strategic coordination in all dyadic common interest games. Previous experimental evidence has provided evidence for strong Stackelberg reasoning in asymmetric games. Here we report evidence from two experiments consistent with players being influenced by strong Stackelberg reasoning in a wide variety of symmetric 3 × 3 games but tending to revert to other choice criteria when strong Stackelberg reasoning generates small payoffs.