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 entry occurs in a positive fraction of the markets. One interpretation is that reputation is not permanent, but must be renewed occasionally.