Optimal Dynamic Auctions Are Virtual Welfare Maximizers

We are interested in the setting where a seller sells sequentially arriving items, one per period, via a dynamic auction. At the beginning of each period, each buyer draws a private valuation for the item to be sold in that period and this valuation is independent across buyers and periods. The auction can be dynamic in the sense that the auction at period t can be conditional on the bids in that period and all previous periods, subject to certain appropriately defined incentive compatible and individually rational conditions. Perhaps not surprisingly, the revenue optimal dynamic auctions are computationally hard to find and existing literatures that aim to approximate the optimal auctions are all based on solving complex dynamic programs. It remains largely open on the structural interpretability of the optimal dynamic auctions. In this paper, we show that any optimal dynamic auction is a virtual welfare maximizer subject to some monotone allocation constraints. In particular, the explicit definition of the virtual value function above arises naturally from the primal-dual analysis by relaxing the monotone constraints. We further develop an ironing technique that gets rid of the monotone allocation constraints. Quite different from Myerson’s ironing approach, our technique is more technically involved due to the interdependence of the virtual value functions across buyers. We nevertheless show that ironing can be done approximately and efficiently, which in turn leads to a Fully Polynomial Time Approximation Scheme of the optimal dynamic auction.

Ex-post IR Dynamic Auctions with Cost-per-Action Payments

Motivated by online ad auctions, we consider a repeated auction between one seller and many buyers, where each buyer only has an estimation of her value in each period until she actually receives the item in that period. The seller is allowed to conduct a dynamic auction but must guarantee ex-post individual rationality. In this paper, we use a structure that we call credit accounts to enable a general reduction from any incentive compatible and ex-ante individual rational dynamic auction to an approximate incentive compatible and ex-post individually rational dynamic auction with credit accounts. Our reduction obtains stronger individual rationality guarantees at the cost of weaker incentive compatibility. Surprisingly, our reduction works without any common knowledge assumption. Finally, as a complement to our reduction, we prove that there is no non-trivial auction that is exactly incentive compatible and ex-post individually rational under this setting.

An Efficient Ascending-Bid Auction for Multiple Objects: Reply

In a comment, Okamoto (2018 ) identifies and corrects a misspecification of the rationing rule in Ausubel (2004 ). This reply elaborates on the observation that the optimality of truthful bidding in dynamic auctions may be sensitive to the fine details of the rationing rule. It then discusses the wider role of sequential bid processing in restoring truthful bidding. (JEL D44)