One item is auctioned to 12 bidders in a sealed-bid auction. The valuation of bidder is denoted by ( = 1,2, … , 12) and is drawn independently from the uniform distribution on [0,1].
a) The auctioneer arranges a second price sealed-bid auction. Show that in a symmetric equilibrium each bidder has no incentive to bid other than truthfully (ie. to bid her own valuation).
b) The auctioneer arranges a first price sealed-bid auction. Without deriving each bidder’s optimal bidding strategy, show that in a symmetric equilibrium each bidder has an incentive to bid lower than her own valuation.
c) The auctioneer arranges a third price sealed-bid auction. Without deriving each bidder’s optimal bidding strategy, show that in a symmetric equilibrium each bidder has an incentive to bid higher than her own valuation.