We develop a sampling theory for games with a large number of players that are exchangeable from the econometrician’s perspective. We show that in the limit, heterogeneity can be separated into an individual and an aggregate component, and establish conditional laws of large numbers
and central limit theorems given the aggregate state of the game. We then develop estimation procedures that eliminate heterogeneity in the aggregate through conditioning and invariance properties, which allow consistent estimation and inference based on information from a finite number of games as the number of players grows large. Estimation is based on the marginal distribution of individual observable characteristics and choices, where we treat the equilibrium selection rule as an unknown nuisance parameter. We illustrate the applicability of our results for aggregative games and matching markets.