We have studied auctions where all bidders are present at the start. There are other online settings (e.g. hotels or airline tickets) where the customers arrive over time. In such settings we can hold an auction where bidders arrive one at a time. Consider selling a single item. We can offer each bidder a price for the item as they arrive. If they are willing to pay the price they buy the item. Otherwise they leave and do not return. The rule for setting these prices could be complex (e.g. depend on the entire history of previous arrivals.

Argue that any such auction is strategyproof.