Repeated trade with imperfect information about previous transactions

This paper studies repeated trade with noisy information about previous transactions. A buyer has private information about his willingness to pay, which is either low or high, and buys goods from different sellers over time. Each seller observes a noisy history of signals about the buyer's previous purchases and sets a price. We compare the cases where previous prices are observable to sellers with the case where they are not. We show that, in both cases, more signal precision is counterbalanced in equilibrium by less separation between types, either because (1) sellers offer discounted prices more often, which are accepted by both types, or (2) the buyer with a high valuation rejects high prices with a higher probability. The effect on welfare and efficiency of making prices observable depends on the signal precision: When the signal is imprecise, making prices public strengthens the discounting mechanism, improving efficiency and buyer welfare; when the signal is precise, making prices public activates the rejection mechanism, and efficiency and buyer welfare may decrease. Independently of price observability, the buyer tends to benefit from a more precise signal about previous purchases.