Problem consider a first-price auction with three bidders

Question

Problem: 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 v_i is below y.

a. Suppose that player 2 is using the bidding function b₂(v₂) = (3/4)v₂, and player 3 is using the bidding function b₃(v₃) = (4/5)v₃. 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 v₁ and her bid b₁.

b. Disregard the assumption made in part (a). Calculate the Bayesian Nash equilibrium of this auction game and report the equilibrium bidding functions.