We study a vertically differentiated duopoly where firms can base their prices on purchase history of customers as in behavior Based Price Discrimination models. However, like in Choe et al. (2018) model for horizontal differentiation, we assume that a firm can perfectly price discriminate customers that have purchased from itself in the previous period, while it can only quote a group price to other customers. We show that the vertical differentiation model retains the one-way poaching in the second period, but is otherwise markedly different from the horizontal one as to the prediction of what are the strategic relevant tools for firms. The main determinant of the equilibrium properties is the extent of what we call “natural market” of each firm, namely the set of consumers that cannot be attracted by an offer made by the rival in the second period (cannot be “poached”). This natural market is determined by cost and quality differences and does not depend upon pricing strategies. The allocation of consumers is never optimal in equilibrium. The firm with the largest natural market is a “fat cat” player and never sells to the whole extent of its natural market, but it makes higher profits than the rival. Consumers gain and firms lose from the use of the customer specific information. Also, we obtain almost everywhere unique equilibria, in sharp contrast to the horizontal case.