Identification and Estimation of Dynamic Games When Players’ Beliefs Are Not in Equilibrium

This paper deals with the identification and estimation of dynamic games when players' beliefs about other players' actions are biased, i.e., beliefs do not represent the probability distribution of the actual behavior of other players conditional on the information available. First, we show that a exclusion restriction, typically used to identify empirical games, provides testable nonparametric restrictions of the null hypothesis of equilibrium beliefs. Second, we prove that this exclusion restriction, together with consistent estimates of beliefs at several points in the support of the special state variable (i.e., the variable involved in the exclusion restriction), is sufficient for nonparametric point-identification of players' payoffs and belief functions. The consistent estimates of beliefs at some points of support may come either from an assumption of unbiased beliefs at these points in the state space, or from available data on elicited beliefs for some values of the state variables. Third, we propose a simple two-step estimation method. We illustrate our model and methods using both Monte Carlo experiments and an empirical application of a dynamic game of store location by retail chains. The key conditions for the identification of beliefs and payoffs in our application are the following: (a) the previous year's network of stores of the competitor does not have a direct effect on the profit of a firm, but the firm's own network of stores at previous year does affect its profit because the existence of sunk entry costs and economies of density in these costs; and (b) firms' beliefs are unbiased in those markets that are close, in a geographic sense, to the opponent's network of stores, though beliefs are unrestricted, and potentially biased, for unexplored markets which are farther away from the competitors' network. Our estimates show significant evidence of biased beliefs. Furthermore, imposing the restriction of unbiased beliefs generates a substantial attenuation bias in the estimate of competition effects.