In a standard English auction participants keep on bidding until one person is left with an unambiguously highest offer.
In a Dutch auction, which is used to sell tulips, the auctioneer starts out with a high price and keeps on lowering it until someone bites. The auction then ends.
A large literature considers the conditions under which these two approaches yield the same expected revenue.
In terms of dating, if you run an English auction you go out with many people, if not simultaneously then relatively closely bunched in time, and you stick with the one who offers the most. If you run a Dutch auction you signal clearly your standards (lowering the standards over time if need be), and stick with the first person who bites.
I believe that strongly Christian women are more likely to run a Dutch auction. Perhaps non-religious women are more likely to run Dutch auctions as they get older.
The economics literature suggests that if bidders are risk-averse, the Dutch auction is more likely to yield higher revenue (in the limiting case, grab quickly any positive consumer surplus, before it goes away). Alternatively, if the seller thinks that bidders would be more enthusiastic if they saw each other’s ongoing bids, this tends to encourage an English auction. In other words, hidden but not too hidden qualities encourage English auctions.
Here are other kinds of auctions.