Price Discrimination Based on Buyers' Purchase History

We consider a repeated sales situation in which a seller sells one unit of a good each period to the same buyer. Having incomplete information about the buyer's valuation, the seller responds to the buyer's behavior, meaning, whether she bought previous units and at what price. We characterize the equilibrium of the no-commitment game for a uniform distribution and show that the seller discriminates between buyers based on their purchase history. Moreover, we show that the buyer's ex-ante expected payoff is higher in the no-commitment game than in a game in which the seller can commit in advance to the prices he will charge (the commitment game). We also demonstrate that the buyer's ex-post payoff is higher for each type of buyer in the no-commitment game. Social welfare is also higher in the no-commitment game. As the number of periods (units) increases, the opening prices decline, but fewer types of buyers accept the opening price.