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

This article deals with the identification and estimation of dynamic games when players’ beliefs about other players’ actions are biased, that is, beliefs do not represent the probability distribution of the actual behaviour of other players conditional on the information available. First, we show that an exclusion restriction, typically used to identify empirical games, provides testable non-parametric restrictions of the null hypothesis of equilibrium beliefs in dynamic games with either finite or infinite horizon. We use this result to construct a simple Likelihood Ratio test of equilibrium beliefs. Second, we prove that this exclusion restriction, together with consistent estimates of beliefs at two points in the support of the variable involved in the exclusion restriction, is sufficient for non-parametric point-identification of players’ belief functions as well as useful functions of payoffs. 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: (1) 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 (2) 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.