Optimal Dynamic Mechanism under Customer Search

This paper investigates the seller’s revenue maximizing mechanism in face of a customer who searches for outside alternatives over a finite horizon. Customer’s utility from searches is modeled as a general function—referred to as the recall function—of the past search outcomes. The seller, without observing the customer’s valuation to her product nor any realization of the search outcomes, can propose a contract to the customer before he begins any searches. Under a general recall function, we show that it is optimal for the seller to offer a menu of American options comprising of a menu of deposits and strike prices, wherein the customer is screened based on the valuation for the seller’s product. When the customer can only recall a few recent outside alternatives, we further establish that under the optimal mechanism, low valuation customer searches for outside alternatives without engaging with the seller while high valuation customer exercises the option immediately—effectively making the option an exploding offer. Customer with intermediate valuation will always hold the option till the end of the planning horizon. While longer search horizon or smaller search cost both increase the customer’s utility from searches, they have different impact on the seller’s revenue. More search opportunities lead the seller’s optimal revenue to decrease exponentially, and in the limit, the optimal mechanism converges to a posted price mechanism. In comparison, as search cost increases, the seller’s revenue may first decrease then increase. In the extreme case when the search cost exceeds the average value of outside alternatives, the customer’s sequential search problem reduces to the one in which he strategically times his purchases of the seller’s product. Our optimal mechanism in this case reduces to making a single exploding offer with monopoly price and hence recovering the classic result in Stokey (1979)