Semiparametric estimation of a simultaneous game with incomplete information

This paper studies the problem of estimating the normal-form payoff parameters of a simultaneous, discrete game where the realization of such payoffs is not common knowledge. The paper contributes to the existing literature in two ways. First, by making a comparison with the complete information case it formally describes a set of conditions under which allowing for private information in payoffs facilitates the identification of various features of the game. Second, focusing on the incomplete information case it presents an estimation procedure based on the equilibrium properties of the game that relies on weak semiparametric assumptions and relatively flexible information structures which allow players to condition their beliefs on signals whose exact distribution function is unknown to the researcher. The proposed estimators recover unobserved beliefs by solving a semiparametric sample analog of the population Bayesian-Nash equilibrium conditions. The asymptotic features of such estimators are studied for the case in which the distribution of unobserved shocks is known and the case in which it is unknown. In both instances equilibrium uniqueness is assumed to hold only in a neighborhood of the true parameter value and for a subset Z of realizations of the signals. Multiple equilibria are allowed elsewhere in the parameter space and no equilibrium selection theory is involved. Extensions to games where beliefs are conditioned on unobservables as well as general games with many players and actions are also discussed. An empirical application of a simple capital investment game is included.