Exercise 1. Consider the following first-price sealed-bid auction. There are two bidders. Each bidder i observes a random variable ,, distributed IID uniformly on the interval [0, 1]. The value of the object to bidder i is given by v 2₁+. Each bidder i submits a sealed bid b; € [0, 00). Bidder i's payoff is given by i-by if b, > by, where ji, (b) if bb;, where ji, 0 otherwise. Find an equilibrium in which each bidder submits a bid of the form B₁(x) = c + di for some constants c and d. In this equilibrium, what is bidder i's expected payoff conditional on ₁ - x? u(b₁,b₂, 01, 0₂):

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