Buy-price English auction

In English auctions, introducing a buy price, i.e., the seller’s maximum price bid at which any bidder at any time can immediately win the auction, allows the seller to gain higher expected utility than that in a traditional auction when either the seller or the buyers are risk-averse. If the seller sets the buy price high enough, the buyprice English auction is efficient and guarantees the highest bidder wins. Under this condition, the expected utilities of uniformly risk-averse buyers with CARA utilities remain the same as in a traditional auction, and the buy-price and traditional auctions are revenue equivalent when both the seller and the buyers are risk neutral. A bidder with a valuation above the buy price follows one of three possible pure strategies: If his valuation is very high, he would use the buy price unconditionally without trying to observe other bidders; If his valuation is relatively high, he would use the buy price on the condition that there exists another competing bidder; Otherwise, he would use the buy price when the current high bid reaches a threshold level between the reserve and the buy price.