We study dynamic mechanisms for optimizing revenue in repeated auctions, that are robust to heterogeneous forward-looking and learning behavior of the buyers. Typically it is assumed that the buyers are either all myopic or are all infinite lookahead, and that buyers understand and trust the mechanism. These assumptions raise the following question: is it possible to design approximately revenue optimal mechanisms when the buyer pool is heterogeneous? Facing a heterogeneous population of buyers with an unknown mixture of @math -lookahead buyers, myopic buyers, no-regret-learners and no-policy-regret learners, we design a simple state-based mechanism that achieves a constant fraction of the optimal achievable revenue.
There is a large body of work on designing mechanisms in repeated interactions when the buyers' values evolve over time. See @cite_28 @cite_2 @cite_17 @cite_25 @cite_23 @cite_5 @cite_36 @cite_10 @cite_11 @cite_18 for an overview of dynamic mechanisms in such settings.
@cite_18 @cite_11 @cite_28 @cite_36 @cite_23 @cite_2 @cite_5 @cite_10 @cite_25 @cite_17
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We study the dynamic mechanism design problem of a seller who repeatedly sells independent items to a buyer with private values. In this setting, the seller could potentially extract the entire buyer surplus by running efficient auctions and charging an upfront participation fee at the beginning of the horizon. In some markets, such as internet advertising, participation fees are not practical since buyers expect to inspect items before purchasing them. This motivates us to study the design of dynamic mechanisms under successively more stringent requirements that capture the implicit business constraints of these markets. We first consider a periodic individual rationality constraint, which limits the mechanism to charge at most the buyer's value in each period. While this prevents large upfront participation fees, the seller can still design mechanisms that spread a participation fee across the first few auctions. These mechanisms have the unappealing feature that they provide close-to-zero buyer utility in earlier auctions in exchange for higher utility in later auctions. To address this problem, we introduce a martingale utility constraint, which imposes the requirement that from the perspective of the buyer, the next item's expected utility is equal to the present one's. Our main result is providing a dynamic auction satisfying martingale utility and periodic individual rationality whose loss in profit with respect to first-best (full extraction of buyer surplus) is optimal up to polylogarithmic factors. The proposed mechanism is a dynamic two-tier auction with a hard floor and a soft floor that allocates the item whenever the buyer's bid is above the hard floor and charges the minimum of the bid and the soft floor. We examine the design of incentive-compatible screening mechanisms for dynamic environ- ments in which the agents'types follow a (possibly non-Markov) stochastic process, decisions may be made over time and may aect the type process, and payos need not be time-separable. We derive a formula for the derivative of an agent's equilibrium payowith respect to his cur- rent type in an incentive-compatible mechanism, which summarizes all …rst-order conditions for incentive compatibility and generalizes Mirrlees's envelope formula of static mechanism design. We provide conditions on the environment under which this formula must hold in any incentive- compatible mechanism. When specialized to quasi-linear environments, this formula yields a dynamic "revenue-equivalence"result and an expression for dynamic virtual surplus, which is instrumental for the design of optimal mechanisms. We also provide some su¢ cient conditions for incentive compatibility, and for its robustness to an agent's observation of the other agents' past and future types. We apply these results to a number of novel settings, including the de- sign of pro…t-maximizing auctions and durable-good selling mechanisms for buyers whose values We study revenue optimization in a repeated auction between a single seller anda single buyer. Traditionally, the design of repeated auctions requires strong modeling assumptions about the bidder behavior, such as it being myopic, infinite lookahead, or some specific form of learning behavior. Is it possible to design mechanisms which are simultaneously optimal against a multitude of possible buyer behaviors? We answer this question by designing a simple state-based mechanism that is simultaneously approximately optimal against a k -lookahead buyer for all k , a buyer who is a no-regret learner, and a buyer who is a policy-regret learner. Against each type of buyer our mechanism attains a constant fraction of the optimal revenue attainable against that type of buyer. We complement our positive results with almost tight impossibility results, showing that the revenue approximation tradeoffs achieved by our mechanism for different lookahead attitudes are near-optimal. We introduce a dynamic mechanism design problem in which the designer wants to offer for sale an item to an agent, and another item to the same agent at some point in the future. The agent's joint distribution of valuations for the two items is known, and the agent knows the valuation for the current item (but not for the one in the future). The designer seeks to maximize expected revenue, and the auction must be deterministic, truthful, and ex post individually rational. The optimum mechanism involves a protocol whereby the seller elicits the buyer's current valuation, and based on the bid makes two take-it-or-leave-it offers, one for now and one for the future. We show that finding the optimum deterministic mechanism in this situation --- arguably the simplest meaningful dynamic mechanism design problem imaginable --- is NP-hard. We also prove several positive results, among them a polynomial linear programming-based algorithm for the optimum randomized auction (even for many bidders and periods), and we show strong separations in revenue between non-adaptive, adaptive, and randomized auctions, even when the valuations in the two periods are uncorrelated. Finally, for the same problem in an environment in which contracts cannot be enforced, and thus perfection of equilibrium is necessary, we show that the optimum randomized mechanism requires multiple rounds of cheap talk-like interactions.