Non-cooperative game theory is at its heart a theory of cognition, specifically a theory of how decisions are made. Game theory's leverage is that we can design different payoffs, settings, player arrays, action possibilities, and information structures, and that these differences lead to different strategies, outcomes, and equilibria. It is well-known that, in experimental settings, people do not adopt the predicted strategies, outcomes, and equilibria. The standard response to this mismatch of prediction and observation is to add various psychological axioms to the game-theoretic framework. Regardless of the differing specific proposals and results, game theory uniformly makes certain cognitive assumptions that seem rarely to be acknowledged, much less interrogated. Indeed, it is not widely understood that game theory is essentially a cognitive theory. Here, we interrogate those cognitive assumptions. We do more than reject specific predictions from specific games. More broadly, we reject the underlying cognitive model implicitly assumed by game theory.