Selling to a No-Regret Buyer

We consider the problem of a single seller repeatedly selling a single item to a single buyer (specifically, the buyer has a value drawn fresh from known distribution $D$ in every round). Prior work assumes that the buyer is fully rational and will perfectly reason about how their bids today affect the seller's decisions tomorrow. In this work we initiate a different direction: the buyer simply runs a no-regret learning algorithm over possible bids. We provide a fairly complete characterization of optimal auctions for the seller in this domain. Specifically: - If the buyer bids according to EXP3 (or any "mean-based" learning algorithm), then the seller can extract expected revenue arbitrarily close to the expected welfare. This auction is independent of the buyer's valuation $D$, but somewhat unnatural as it is sometimes in the buyer's interest to overbid. - There exists a learning algorithm $\mathcal{A}$ such that if the buyer bids according to $\mathcal{A}$ then the optimal strategy for the seller is simply to post the Myerson reserve for $D$ every round. - If the buyer bids according to EXP3 (or any "mean-based" learning algorithm), but the seller is restricted to "natural" auction formats where overbidding is dominated (e.g. Generalized First-Price or Generalized Second-Price), then the optimal strategy for the seller is a pay-your-bid format with decreasing reserves over time. Moreover, the seller's optimal achievable revenue is characterized by a linear program, and can be unboundedly better than the best truthful auction yet simultaneously unboundedly worse than the expected welfare

Robust Repeated Auctions under Heterogeneous Buyer Behavior

We study revenue optimization in a repeated auction between a single seller and a 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