Reputation and exogenous private learning

I consider a modified version of the chain store game in which the short-run competitors receive noisy signals of the long-run incumbent firm’s type. The history of signals is observable to the competitors but not to the incumbent. There are two main results. First, for any p ∈ (0, 1) there exists a T < ∞ such that within T periods the competitors assign belief p to the incumbent’s realized type with probability at least p in equilibrium, regardless of the discount factor. Second, as long as there is sufficient noise in the entrant’s signals, then in any equilibrium the weak type of incumbent obtains a payoff strictly higher than her minmax payoff if she is patient enough.