Auction

Types of Auction  Open outcry English (ascending) auction Dutch (descending) auction  Sealed bid First-price Second-price (Vickrey)  Equivalence in these auctions?  Common-value auction  Private-value auction

 The winner ’ s curse When one places a bid (say $b) in a common-value auction and is accepted, the value to the seller ’ s must be less than $b (0-b). Otherwise the item will not be sold. The common value is probably about $b/2 given the average is an unbiased estimate. (Note: everyone ’ s estimate is private information.)

Auction and incomplete information  In the private-value auction, every bidder has private information about her evaluation on the item sold. Ex: vi~U(0,1),i.i.d. bidders are risk neutral

Bidding Strategy  First-price Bidder with higher valuation will bid higher Given your evaluation is highest, expectation of the 2nd highest valuation Ex: if one ’ s valuation is x, then bids (n-1)x/n

 Second-price The bidder will bid truthfully her evaluation on the item. Truthfully bidding is a weakly dominant strategy.  Revenue Equivalence Principle The seller will collect the same amount from either 1 st -price of 2 nd -price auction. Assumptions of independence, identical distribution, risk neutral matters.

 All-pay auction Every bidder pays a sunk (unrecoverable) cost of her bid, and the one with the highest bid wins the item. Ex: Olympic games, political lobbying, R&D races  Equilibrium bidding strategy must be a mixed strategy.  Consider a common-value all-pay auction with prize worth 1.

 Bid x in (0,1) will be continuous  Let P(x) be the equilibrium cdf, the probability one ’ s bid is not higher than x.  Indifference principle  Bidding x: [P(x)] n-1 -x  Bidding 0: 0  → P(x)=x 1/(n-1)  When n=2, P(x)=x A uniform distribution for pdf of x

 When n=3, P(x)=x 1/2 X=1/4, P(x)=1/2, ½ of chance the bid will be lower than 1/4  As n increases, it ’ s more likely to bid lower.  Expected payment is 1/n for everyone.