You are hired to study a particular auction in which a single indivisible good is for sale. You find out that N bidders participate in the auction and each has a private value v ∈ [0, 1] drawn independently from the common density f(v) = 2v, whose cumulative distribution function is F(v) = v2. All you know about the auction rules is that the highest bidder wins. But you do not know what he must pay, or whether bidders who lose must pay as well. On the other hand, you do know that there is an equilibrium, and that in equilibrium each bidder employs the same strictly increasing bidding function (the exact function you do not know), and that no bidder ever pays more than his bid.

(a) Which bidder will win this auction?

(b) Prove that when a bidder’s value is zero, he pays zero and wins the good with probability zero.

(c) Using parts (a) and (b), prove that the seller’s expected revenue must be 

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