Identification of games of incomplete information with multiple equilibria and unobserved heterogeneity

This paper deals with identification of discrete games of incomplete information when we allow for three types of unobservables: payoff‐relevant variables, both players' private information and common knowledge, and nonpayoff‐relevant variables that determine the selection between multiple equilibria. The specification of the payoff function and the distributions of the common knowledge unobservables is nonparametric with finite support (i.e., finite mixture model). We provide necessary and sufficient conditions for the identification of all the primitives of the model. Two types of conditions play a key role in our identification results: independence between players' private information, and an exclusion restriction in the payoff function. When using a sequential identification approach, we find that the up‐to‐label‐swapping identification of the finite mixture model in the first step creates a problem in the identification of the payoff function in the second step: unobserved types have to be correctly matched across different values of observable explanatory variables. We show that this matching‐type problem appears in the sequential estimation of other structural models with nonparametric finite mixtures. We derive necessary and sufficient conditions for identification, and show that additive separability of unobserved heterogeneity in the payoff function is a sufficient condition to deal with this problem. We also compare sequential and joint identification approaches.