Structural estimation of discrete-choice games of incomplete information with multiple equilibria

Abstract

Estimation of games with multiple equilibria has received much attention in the recent econometrics literature. Unlike other estimation problems such as single-agent dynamic decision models or demand estimation, in which there is a unique solution in the underlying structural models, games usually admit multiple equilibria and the number of equilibria in a game can vary for different structural parameters. This fact makes the estimation of games far more challenging because the likelihood function or other criterion function defined in the space of structural parameters can be discontinuous or non-differentiable. Two-step estimators by Bajari et al. (2007) and Pesendorfer and Schmidt-Dengler (2008) and Nested Pusedo Likelihood (NPL) estimators by Aguirregabiria and Mira (2007) are proposed to address this problem. We recast the estimation problem as a constrained optimization problem with the Bayesian-Nash equilibrium condition being the constraints. The advantage of our formulation is that the likelihood function, now defined in the equilibrium probability space, is continuous and smooth. This allows researchers to use state-of-the-art optimization software to solve the estimation problem. In a Monte Carlo study, we compare the performance of a two-step estimator, NLP estimator, and our constrained optimization estimator.