Economists have been investigating the link between institutions and economic performance for several years. While econometric studies of this link have flourished, they are of limited use in understanding the causal mechanisms making some institutions responsible for economic performance. Several works using game theory and akin to the ‘new institutional economics’ have entertained the goal of developing micro-explanations of the institutions–performance link. Because game theory focuses on individuals’ actions and beliefs, game-theoretic studies of institutions are thought to oppose more ‘structuralist’ explanations that downplay the role of individual agents and put more emphasis on the importance of social (or ‘macro’) structures. This paper demonstrates that this claim is misconceived, as the micro-explanations produced by game-theoretic models must assume already existing macro-structures. Institutions produce downward effects, shaping each agent's action. Moreover, in a game-theoretic framework, macro-structures are constitutive of individual agency since, without them, agents would often be unable to choose. I illustrate this claim with the example of Avner Greif's study of the role of cultural beliefs in the economic organisation of medieval societies.