The Trouble With First-Price Auctions

While modeling first-price auctions is probably more useful than modeling second-price auctions, it is unfortunately also quite a bit more difficult mathematically. While in the second-price case, the highest bid sometimes has little to do with the price paid, in the first-price case it is exactly the price paid. This means that bidding the expected value of the item (which is often an optimal strategy in the second-price case) can only lead to zero or negative expected profits in the first-price case.