(First-price procurement auction) We are used to auctions in which there is a single seller facing many potential buyers (i.e., the bidders). However, in many situations we have a single buyer who wants to buy a good (or service) from one of many sellers. In these situations, the buyer may want to run an auction. These are called procurement (or reverse) auctions and are very common both in the public and private world. In a procurement auction, sellers bid for the price at which they are willing to sell their item (or service). In this context, it is natural to talk about a bidder's cost (to provide a service or good) instead of a bidder's valuation. Let's consider a concrete example. Suppose your parents want to remodel their kitchen. They call two contractors (A and B) and ask them for quotes. Your parents tell them that they are going to run a sealed-bid first-price auction. That is, the contractors should email them their quotes, the contractor with the lowest quote wins, and gets paid her quote. Suppose that the contractors are symmetric and have private costs, and that their costs are independently drawn from a Uniform [0, 1] distribution. In this simple 2-bidder procurement auction, the bidding strategy 3*(c) = (1 + c) is a (symmetric) Bayesian Nash Equilibrium. That is, in equilibrium, it is optimal for a contractor who anticipates a cost of completing the project of c to submit a quote equal to (1+c). (a) If bidders follow the equilibrium bidding strategy, do they bid above or below their cost? How do you interpret the difference between a contractor's quote and their cost? (b) Write down the expression for the probability of contractor A winning when she submits a quote equal to b and contractor B submits a quote according to the equilibrium bidding function 8*(). (c) Write down the expression for contractor A's expected payoff when she has a cost of completing the project equal to c and submits a quote equal to b, and her rival bids according to the equilibrium strategy 8*(-). (d) Show that the bidding strategy 3*(c)=(1+c) is in fact a (symmetric) Bayesian Nash Equilibrium.