Consider a first-price auction with three bidders, whose valuations are independently drawn from a uniform distribution on the interval [0, 30]. Thus, for each player i and any fixed number y ∈ [0, 30], y>30 is the probability that player i's valuation vi is below y.
(a) Suppose that player 2 is using the bidding function b2(v2) = (3/)
2 , and player 3 is using the bidding function b3(3) = (4/5)3 . Determine player 1's optimal bidding function in response. Start by writing player 1's expected payoff as a function of player 1's valuation 1 and her bid b1 .
(b) Disregard the assumptions made in part (a). Calculate the Bayesian Nash equilibrium of this auction game and report the equilibrium bidding functions.