Peculiar Revenue-Equivalent Auctions
The Revenue Equivalence Theorem says that a symmetric independent private value auction has the same revenue regardless of which auction rule is used, so long as the auction rule awards the object to the highest-valuing buyer (which rules out reserve prices), and a buyer with value zero has a zero payoff (which rules out entry fees). The four basic auction rules are the Ascending, Descending, First-Price, and Second-Price auctions. The All-Pay auction is now standardly put on the basic list too. I’ll list some others here.
First, here is the revenue equivalence theorem:
THE REVENUE EQUIVALENCE THEOREM. Let all players be risk-neutral with private values drawn independently from the same atomless, strictly increasing distribution F(v) on [v0, vt]. If under either Auction Rule A1 or Auction Rule A2 it is true that:
(a) The winner of the object is the player with the highest value;
(b) The lowest bidder type, v = v0, has an expected payment of zero
then the symmetric equilibria of the two auction rules have the same expected payoffs for each type of bidder and for the seller.
(1) The Losers Pay Double Auction.
The winner is whoever bids the highest. He pays his bid, and the losers each pay double their bids. This is a variant of the all-pay auction.
(2) The Losers Pay Auction.
The winner is whoever bids the highest. He pays 1/1000 of his bid, and the losers each pay their bids. Note that we need for the winner to pay something, and for that something to be proportional to his bid, or there is no equilibrium, because each bidder will want to bid higher than the other bidders and does not care how much higher (the Dollar Auction of Shubik).
(3) The Third-Price Auction.
The winner is whoever bids the highest. He pays the amount bid by the third-highest bidder. Krishna discusses this on pages 32-34 of his book, Auction Theory. The Lowest-Price Auction is similar to this. Note that it is not dominant to bid an infinite amount, because if the third or lowest price is high enough, a bidder would regret winning even at that price.
(4) The Average-Price Auction.
The winner is whoever bids the highest. He pays the average amount bid, across all bidders. I think the U.S. Treasury has used something similar to this for selling T-bills, but since they are selling multiple items, using the average price across all the successful bids.
(5) The (i+1)th-Price All-Pay Auction.
The winner is whoever bids the highest. He pays the amount of the second highest bid. The second highest-bidder pays the amount of the third-highest bid. The i’th highest bidder pays the amount of the (i+1)th highest bid. The lowest bidder pays zero.