n this problem you will find the optimal strategy for the buyer and revenue to the seller in a symmetric first-price auction with three bidders, when the private values are independent and identically distributed as U [0, 1]. Recall that in a first-price auction, the highest bidder wins, and pays their bid. If it is symmetric, then i = for all i. You may assume that is strictly increasing on [0, 1] and that all the functions you will use are differentiable. (3 points each) (a) Let's look at the problem from the point of view of player 1, whose private value is v1. Suppose that P1 is going to bid as though their private value is w. Show that their allocation probability a1(w) = w2. (a1(w) is the probability that player 1 wins bidding (w) when the other players are bidding (Vj ) for j = i). (b) As we did in class, we can write the expected utility for player 1, u1[b1] in terms of w: u1[b1|v1] = v1 a1[b1] p1[b1] u1(w|v1) = v1 a1(w) p1(w) Use this to show that p1(v1) = 2v3 1 /3 (c) Use the expected