Estimation of games with ordered actions: An application to chain‐store entry

We study the estimation of static games where players are allowed to have ordered actions, such as the number of stores to enter into a market. Assuming that payoff functions satisfy general shape restrictions, we show that equilibrium of the game implies a covariance restriction between each player's action and a component of the player's payoff function that we call the strategic index. The strategic index captures the direction of strategic interaction (i.e., patterns of substitutability or complementarity) as well as the relative effects of opponents' decisions on players' payoffs. The covariance restriction we derive is robust to the presence of multiple equilibria, and provides a basis for identification and estimation of the strategic index. We introduce an econometric method for inference in our model that exploits the information in moment inequalities in a computationally simple way. We analyze its properties through Monte Carlo experiments and then apply our approach to study entry behavior by chain stores where there is both an intensive margin of entry (how many stores to open in a market) as well as the usual extensive margin of entry (whether to enter a market or not). Using data from retail pharmacies we find evidence of asymmetries in strategic effects among firms in the industry that has implications for merger policy. We also find that business stealing effects are less pronounced in larger markets, which helps explain the large positive correlation in entry behavior observed in the data.