3 NE in Auctions (20 points) Consider a rstrprioe sealed bid auction with n riskneutral bidders [we've been examining risk neutral bidders all along]. Each bidder has a private value independently drawn from a uniform distribution on {[1,1]. That is, for each bidder, all values between i] and 1 are equally likely. The complete strategy of each bidder is a bid function that will tell us, for any value 1:, what amount b(u} that bidder will choose to bid. It is proposed that the equilibrium bid function for n. = 2 is Mo} = uf2 for each of the two bidders. That is, ifwe have two bidders, each should bid half her value. a} (4 points) Suppose you're bidding against just one opponent whose value is univormly distributed on 10,1] and always bids half her value. What is the probability that you will win if you bid b=.1? If you bid b=.4'? If you bid b=.? b} (:1 points} Put together the answers to part a}. What is the correct mathematical expression for p(win], the probability that you win, as a function of your bid .5? c} (4 points) Find an expression for the expected prot you make when your value is u and you bid is t, given that your opponent is bidding half her value. Member that there are two cases: either you win the auction or you lose the auction. You need to average the prot between these cases. d} (4 points} What is the value of b that maximizes your expected prot? This should be a function of your value 1:. e} (4 points] Use your results to argue that it is a NE for both biddres to follow the same bid function 33(1):]: = 1.12