Inference in games without Nash equilibrium: An application to restaurants' competition in opening hours

This paper relaxes the Bayesian Nash equilibrium (BNE) assumption commonly imposed in empirical discrete choice games with incomplete information. Instead of assuming that players have unbiased/correct expectations, my model treats a player's belief about the behavior of other players as an unrestricted unknown function. I study the joint identification of belief and payoff functions. I show that in games where one player has more actions than the other player, the payoff function is partially identified with neither equilibrium restrictions nor the usual exclusion restrictions. Furthermore, if the cardinality of players' action sets varies across games, then the payoff and belief functions are point identified up to scale normalizations and the restriction of equilibrium beliefs is testable. For games where action sets are constant across players and observations, I obtain very similar identification results without imposing restrictions on beliefs, as long as the payoff function satisfies a condition of multiplicative separability. I apply this model and its identification results to study the store hours competition between McDonald's and Kentucky Fried Chicken (KFC) in China. The null hypothesis that KFC has unbiased beliefs is rejected. Failing to account for KFC's biased beliefs generates an attenuation bias on estimated strategic effects. Finally, the estimation results of the payoff functions indicate that the decision about store hours is a type of vertical differentiation. By operating through the night, a firm not only attracts night-time consumers but also can steal competitors' day-time customers. This result has implications on the optimal regulation of stores' opening hours.