Estimating Simultaneous Games with Incomplete Information Under Median Restrictions

I estimate a simultaneous discrete game with incomplete information where players' private information are only required to be median independent of observed states and can be correlated with observable states. This median restriction is weaker than other assumptions on players' private information in the literature (e.g. perfect knowledge of its distribution or its independence of the observable states). I show index coefficients in players' utility functions are point-identified under an exclusion restriction and fairly weak conditions on the support of states. This identification strategy is fundamentally different from that in a single-agent binary response models with median restrictions, and does not involve any parametric assumption on equilibrium selection in the presence of multiple Bayesian Nash equilibria. I then propose a two-step extreme estimator for the linear coefficients, and prove its consistency