one's true valuation is dominated by the strategy of truthfully bidding. The strategy of underbidding is dominated by bidding truthfully. Assume that bidder i bids . If then the bidder would lose the item with a truthful bid as well as an underbid, so the strategies have equal payoffs for this case. If then the bidder would win the item either way so the strategies have equal payoffs in this case. If then only the strategy of truthfully bidding would win the auction. The payoff for the truthful strategy would be positive as they paid less than their value of the item, while the payoff for an underbid bid would be zero. Thus the strategy of underbidding is dominated by the strategy of truthfully bidding. Truthful bidding dominates the other possible strategies (underbidding and overbidding) so it is an optimal strategy. Revenue equivalence of the Vickrey auction and sealed first price auction The two most common auctions are the sealed first price (or high-bid) auction and the open ascending price (or English) auction. In the former each buyer submits a sealed bid. The high bidder is awarded the item and pays his or her bid. In the latter, the auctioneer announces successively higher asking prices and continues until no one is willing to accept a higher price. Suppose that a buyer's valuation is and the current asking price is . If , then the buyer loses by raising his hand. If and the buyer is not the current high bidder, it is more profitable to bid than to let someone else be the winner. Thus it is a dominant strategy for a buyer to drop out of the bidding when the asking price reaches his or her valuation. Thus, just as in the Vickrey sealed second price auction, the price paid by the buyer with the highest valuation is equal to the second highest value. Consider then the expected payment in the sealed second-price auction. Vickrey considered the case of two buyers and assumed that each buyer's value was an independent draw from a uniform distribution with support . With buyers bidding according to their dominant strategies, a buyer with valuation wins if his opponent's value . Suppose that is the high value. Then the winning payment is uniformly distributed on the interval and so the expected payment of the winner is . We now argue that in the sealed