How should one sell an item to a buyer whose value for the item will only be realized next week? E.g. consider selling a flight to some executive who may or may not have a meeting with a client next week. Suppose that both the seller and the buyer only know a distribution, F , from which the buyer's value, v, for the item will be drawn. One way the seller could go about this sale is to make a take-it-or-leave-it offer today. The offer reads "pay E [v] today to get the item next week." A risk-neutral buyer would find this offer attractive, hence the seller would extract the full surplus, E [v]. The unsettling feature of the afore-described mechanism is that, for some realizations of v, the bidder ends up with negative utility. In particular, while our mechanism is interim Individually Rational (IR), it is not ex-post IR. How could we fix this? One way is to wait until next week when the value is realized and make a take-it-or-leave-it offer of the item at an optimal monopoly price, a price p maximizing p · (1 − F (p)). The new mechanism is clearly ex-post IR, but its revenue could be much smaller than that of the previous one. Still, this trivial mechanism extracts the best possible revenue among all ex-post IR mechanisms, as a simple argument can establish. However, this optimality argument fails when several items are to be sold over consecutive periods. In this paper, we provide a characterization of the revenue-optimal, ex-post IR, dynamic mechanism selling k items over k periods to a bidder whose values are independent. In particular, we optimize the seller's revenue subject to the following strong individual rationality condition: at each period, the stage utility of the agent, defined to be the surplus from that period's allocation minus the agent's payment, must be nonnegative. In particular, the non-negativity of the stage utilities implies that, at the end of each period, the agent's realized utility from participating in the mechanism so far is non-negative. We provide extensions to multiple bidders and an infinite horizon with discount factors. Results. Our characterization reveals structural properties of optimal mechanisms. We show that there exists an optimal mechanism in which, in all periods except for possibly the last, the stage utility of all realized types of the agent is zero; that is, every type is asked to pay its surplus from the allocation. To satisfy incentive compatibility, the mechanism maintains a scalar state variable that dictates the future allocations and payments. More precisely, we show that an optimal mechanism can be described by two functions that depend only on the current state of the mechanism and the agent's bid (rather than the full history of bids): an allocation function
Ashlagi et al. REF provide characterizations of the optimal ex-post IR, periodic IC dynamic mechanism, with m independent stages and n buyers.
Sequential Mechanisms with Ex-post Participation Guarantees
EC '16
Mathematics Economics Computer Science