Consider a sealed-bid first-price auction with two buyers whose private values are independent; the private value of buyer 1 has uniform distribution over the interval [0, 3], and the private value of buyer 2 has uniform distribution over the interval [3, 4]. Answer the following questions:
(a) Prove that the following pair of strategies form an equilibrium

β_{1 (}v_{1) =} 1 + (v₁ / 2),

β2 ₍v_{2) =} (1 / 2₎ + ₍v₂ / 2),

(b) Is the probability that buyer 2 wins the auction equal to 1?

(c) Compute the seller's expected revenue if the buyers implement the strategies (β₁, β₂).

(d) Compute the seller's expected revenue in a sealed-bid second-price auction. Is that the same expected revenue as in part (c) above?