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Suppose there are just two bidders. In a second-price, all-pay auction, the two bidders simultaneously submit sealed bids. The highest bid wins the object and both bidders pay the second-highest bid.

Suppose there are just two bidders. In a second-price, all-pay auction, the two bidders simultaneously submit sealed bids. The highest bid wins the object and both bidders pay the second-highest bid.

(a) Find the unique symmetric equilibrium bidding function. Interpret.

(b) Do bidders bid higher or lower than in a first-price, all-pay auction?

(c) Find an expression for the seller’s expected revenue.

(d) Both with and without using the revenue equivalence theorem, show that the seller’s expected revenue is the same as in a first-price auction.

Consider the following variant of a first-price auction. Sealed bids are collected. The highest bidder pays his bid but receives the object only if the outcome of the toss of a fair coin is heads. If the outcome is tails, the seller keeps the object and the high bidder’s bid. Assume bidder symmetry.

Consider the following variant of a first-price auction. Sealed bids are collected. The highest bidder pays his bid but receives the object only if the outcome of the toss of a fair coin is heads. If the outcome is tails, the seller keeps the object and the high bidder’s bid. Assume bidder symmetry.

(a) Find the unique symmetric equilibrium bidding function. Interpret.

(b) Do bidders bid higher or lower than in a first-price auction?

(c) Find an expression for the seller’s expected revenue.

(d) Both with and without using the revenue equivalence theorem, show that the seller’s expected revenue is exactly half that of a standard first-price auction.

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