Counterfactual Analysis in Empirical Games

Abstract: We address counterfactual analysis in empirical models of games with partially identified parameters, and multiple equilibria and/or randomized strategies, by constructing and analyzing the counterfactual predictive distribution set (CPDS). This framework accommodates various outcomes of interest, including behavioral and welfare outcomes. It allows a variety of changes to the environment to generate the counterfactual, including modifications of the utility functions, the distribution of utility determinants, the number of decision makers, and the solution concept. We use a Bayesian approach to summarize statistical uncertainty. We establish conditions under which the population CPDS is sharp from the point of view of identification. We also establish conditions under which the posterior CPDS is consistent if the posterior distribution for the underlying model parameter is consistent. Consequently, our results can be employed to conduct counterfactual analysis after a preliminary step of identifying and estimating the underlying model parameter based on the existing literature. Our consistency results involve the development of a new general theory for Bayesian consistency of posterior distributions for mappings of sets. Although we primarily focus on a model of a strategic game, our approach is applicable to other structural models with similar features.