  This paper studies the rationalization and identification of binary games
where players have correlated private types. Allowing for correlation is
crucial in global games and in models with social interactions as it represents
correlated information and homophily, respectively. Our approach is fully
nonparametric in the joint distribution of types and the strategic effects in
the payoffs. First, under monotone pure Bayesian Nash Equilibrium strategy, we
characterize all the restrictions if any on the distribution of players'
choices imposed by the game-theoretic model as well as restrictions associated
with two assumptions frequently made in the empirical analysis of discrete
games. Namely, we consider exogeneity of payoff shifters relative to private
information, and mutual independence of private information given payoff
shifters. Second, we study the nonparametric identification of the payoff
functions and types distribution. We show that the model with exogenous payoff
shifters is fully identified up to a single location--scale normalization under
some exclusion restrictions and rank conditions. Third, we discuss partial
identification under weaker conditions and multiple equilibria. Lastly, we
briefly point out the implications of our results for model testing and
estimation.
