Keep ’Em Coming: Detecting and Nurturing Loyalty

We consider a continuous-time game between a buyer and a seller. The buyer privately knows how often he needs to trade. When he does, he can choose to either engage with the seller, who chooses what utility to supply, or search for an alternative. Because time is informative, the seller learns and adjusts her behaviour over time. Without commitment, in the Markov perfect equilibrium, the seller starts with a pooling offer, before experimenting with occasional separating offers. Her payoff is non-monotone –in fact, quasi-convex– in her belief about the buyer’s type. With commitment, the seller can take advantage of limited-time offers to extract all the buyer’s surplus.