Information sharing, personalized pricing, and collusion

We study collusion sustainability in an infinitely repeated game in which firms might price discriminate, by offering personalized prices for the share of consumers they have information about. We do not impose any restrictions to the distribution of consumers and the product characteristic space. In such a general framework we show that when firms share their personal information about consumers, collusion is more difficult to sustain. We also show that, for intermediate levels of the discount factor, an antitrust policy aiming to discourage joint profit maximization and to maximize the consumer surplus should allow information sharing between firms. Instead, a ban on information sharing is optimal only if firms have imperfect information about their own consumers.