Sequential screening

We present a model of price discrimination where a monopolist faces a consumer who is privately informed about the distribution of his valuation for an indivisible unit of good but has yet to learn privately the actual valuation. The monopolist sequentially screens the consumer with a menu of contracts: the consumer self-selects once by choosing a contract and then self-selects again when he learns the actual valuation. A deterministic sequential mechanism is a menu of refund contracts, each consisting of an advance payment and a refund amount in case of no consumption, but sequential mechanisms may involve randomization. We characterize the optimal sequential mechanism when some consumer types are more eager in the sense of first-order stochastic dominance, and when some types face greater valuation uncertainty in the sense of mean-preserving-spread. We show that it can be optimal to subsidize consumer types with smaller valuation uncertainty (through low refund, as in airplane ticket pricing) in order to reduce the rent to those with greater uncertainty. The size of distortion depends both on the type distribution and on how informative the consumer's initial private knowledge is about his valuation, but not on how much he initially knows about the valuation per se.