  We study the design of auctions with dynamic scoring, which allocate a single
item according to a given scoring rule. We are motivated by online advertising
auctions when users interact with a platform over the course of a session. The
platform ranks ads based on a combination of bids and quality scores, and
updates the quality scores throughout the session based on the user's online
activity. The platform must decide when to show an ad during the session. By
delaying the auction, the auctioneer acquires information about an ad's
quality, improving her chances of selecting a high quality ad. However
information is costly, because delay reduces market thickness and in turn
revenue. When should the auctioneer allocate the impression to balance these
forces?
  We develop a theoretical model to study the effect of market design on the
trade-off between market thickness and information. In particular, we focus on
first- and second-price auctions. The auctioneer can commit to the auction
format, but not to its timing: her decision can thus be cast as a real options
problem. We show that under optimal stopping the first-price auction allocates
efficiently but with delay. Instead, the second-price auction generates more
revenue by avoiding delay. The auctioneer benefits from introducing reserve
prices, more so in a first-price auction.
