Three people are bidding on a good in a first-price auction setting (the highest bidder wins and pays their bid). The true value of the good is v = 10 and common to all three people. No one knows the true value of the good, but each person receives a signal of the true value. In particular, person 1 gets a signal s1 = 9, person 2 gets a signal s2 = 10, and person 3 gets a signal s3 = 11.

a) If each person submits a bid equal to their signal, who wins? What is the winner's payoff? What do we call this outcome?

b) Assume that everyone knows that together they received the signals v1, v, and v+1, but they still don't know v = 10. (For example, when person 1 receives s1 = 9 they learn that the true value v must be one of these: 8, 9, or 10. The true value is actually 10, but person 1 did not know if their signal was low, accurate, or high!) Everyone wants to submit the highest bid possible while avoiding any chance of bidding above the true value. What is the bidding strategy that achieves this outcome?

c) If everyone uses the bidding strategy from b), who wins? What is the winner's payoff?