Optimal Dynamic Mechanism Under Customer Search

This paper investigates the seller's revenue -maximizing mechanism in the face of a customer who searches for outside alternatives over a finite horizon. The customer's utility from searches is modeled as a general function-referred to as the recall function-of the past search outcomes. Without observing the customer's valuation of the product or any realization of search outcomes, the seller can propose and commit to a contract with the customer before the search process begins. Under a general recall function, we show that the optimal strategy for the seller is to offer a menu of American options consisting of deposits and strike prices. In the case in which the customer can only recall a few recent outside alternatives, we further establish that, under the optimal mechanism, customers with low valuation search for outside alternatives without engaging with the seller, whereas high -valuation customers exercise the option immediately, effectively turning the option into an exploding offer. Customers with intermediate valuation only exercise the option, if ever, at the end of the search horizon. Whereas a longer search horizon or smaller search cost both increase the customer's utility from searches, they have different impacts on the seller's revenue. More search opportunities lead to an exponential decrease in the seller's revenue, and in the limit, the optimal mechanism converges to a posted price mechanism. In contrast, as the search cost increases, the seller's revenue may initially decrease and then increase. In the extreme case in which the search cost exceeds the average value of outside alternatives, the customer's sequential search problem reduces to strategically timing purchases of the seller's product. Our optimal mechanism, in this case, reduces to making a single exploding offer with a monopoly price.