Personalized pricing with imperfect customer recognition

We consider a duopoly model where firms can identify only a share of consumers, which is positively correlated with the consumer' preferences. Firms charge personalized prices to the consumers they can recognize and a uniform price to the rest of consumers. The firms' available information is given by the combination of two factors: the intensive margin, which determines the share of consumers the firms can recognize in each single location, and the extensive margin, which determines how many locations the firms can identify. Different market configurations emerge according to the size of these margins. We characterize the values of the intensive and extensive margins that maximize firms' profits, and we show that profits are non-monotonic. We also show that the composition, in addition to the size, of the available information - i.e., the mix of these margins - affects firms' profits significantly. Implications for regulatory policies concerning the protection of consumers' information are finally discussed.