The Value of Observing the Buyers' Arrival Time in Dynamic Pricing

We consider a dynamic pricing problem in which a firm sells one item to a single buyer to maximize expected revenue. The firm commits to a price function over an infinite horizon. The buyer arrives at some random time with a private value for the item. He is more impatient than the seller and strategizes over the timing of the purchase in order to maximize his expected utility, which implies either buying immediately, waiting to benefit from a lower price, or not buying. We study the value of the seller's ability to observe the buyer's arrival time in terms of her expected revenue. When the seller can observe the buyer's arrival, she can make the price function contingent on the buyer's arrival time. On the contrary, when the seller can't, her price function is fixed at time zero for the whole horizon. The value of observability (VO) is defined as the worst-case ratio between the expected revenue of the seller when she observes the buyer's arrival and that when she does not. First, we show that, for the particular case in which the buyer's valuation follows a monotone hazard rate distribution, the upper bound of VO is exp(1). Next, we show our main result: in a setting very general on valuation and arrival time distributions: VO is at most 4.911. To obtain this bound, we fully characterize the solution to the observable arrival problem and use this solution to construct a random and periodic price function for the unobservable case. Finally, we show by solving a particular example to optimality that VO has a lower bound of 1.136.