4. (6 points) Consider the auction model with a continuum of possible valuations. Bidder #'s valuation, V;, is drawn from the uniform distri- bution on [0, 1], for i = 1,2,...,n. In other words, the cdf of V;, can be defined as F'(v) = v for v [0, 1] (and, of course, F(v) =0 forv 1). Each bidder's valuation is independent of any other bidder's valuation. Consider the first-price auction. As I have ar- gued in class, the strategy profile in which B;(v) = B(v) = (n1)-v for all v [0,1] and = 1,2,...,n is a Nash equilibrium. For this problem, focus on the case n = 3. (a) Consider Bidder 1. Given Bidders 2 and 3 bid B(v) = 2v/3 for ~ all v [0,1], show that when V7 = 3/4, the best response for bidder 1 to bid 3 2 3 1 Bl-]==--==. 4 3 4 2 Hint: Express his payoff as a function of his bid, b, and show that b = 1/2 maximizes his expected payoff. Suppose the seller uses a posted price p. What is her expected revenue? Which price maximizes her expected revenue? Hint: What is the probability of at least one buyer is willing to pay p? Recall that in the first price auction, the seller's expected revenue is (n1)/(n+1). Compare the seller's revenue from the first-price auction and that from posted-price selling