Consider an all-pay auction with independent private values, with two bidders i = 1, 2. The auction is very similar to that of the first-price auction that we discussed in class, except that each bidder is required to pay their bid, no matter whether they win or not. The valuations of the two players are iid (independently, identically distributed) and, for simplicity, assume they are distributed uniformly U[0, 1].
(a) Show that the symmetric strategies b1(v) = b2(v) = (1/2)v^2 are a Bayesian Nash equilibrium: Compute the expected revenue of the auctioneer in this equilibrium.
(b) Does this auction satisfy the conditions of the Revenue Equivalence Theorem? If yes, explain why and compare the expected revenue the auctioneer obtains in this all-pay auction with that of the first-price and second-price auctions. If no, explain.