An all-pay auction is an auction where bidders bid simultaneously, the highest bidder wins the auction and all bidders have to pay their bids. The goal of this exercise is to find the symmetric BNE of such an auction. Assume there are n bidders with iid valuations that are uniform on [0, 1]. Assume that the symmetric BNE is linear and of the form b₁ = B(v,)" where B > 0 is a constant you need to find.
(a) Given that all the other players use the symmetric linear strategy, find the expected utility for bidder i when her type is u, and she bids b, € [0, B].
(b) Argue first that optimal b, B. Using this an your answer in a) find the optimal bid for player i. Using the fact that the equilibrium is symmetric, find B.