Reputation and Impermanent Types

I consider a version of the chain store game where the incumbent firm’s type evolves according to a Markov process with two states: a “tough†type who always fights entry, and a “weak†type who prefers to accommodate. There exists a minimal level of persistence necessary for the incumbent to be able to sustain any reputation for being tough. Above that level, as the number of markets T increases, in equilibrium play alternates between intervals of entry by competitors and intervals of deterrence. When T is infinite, then regardless of the discount factor there exists a sequential equilibrium in which the incumbent’s payoff is bounded below her Stackelberg payoff. Both results are in contrast to the outcome when the incumbent’s type is fixed. One interpretation is that reputation is not permanent, but must be renewed occasionally