Consider the equilibrium in which every buyer bids his

Question

Consider a sealed-bid second-price auction with two buyers, whose private values are independent; buyer 1's private value is uniformly distributed over [0, 1], and buyer 2's private value is uniformly distributed over [0, 2].

(a) For each buyer, find all weakly dominant strategies.

(b) Consider the equilibrium in which every buyer bids his private value. What is the probability that buyer 1 wins the auction, under this equilibrium? What is the seller's expected revenue in this case?

(c) Prove that at any equilibrium β = (β₁, β₂) satisfying β₁(v) = β₂(v) for all v ∈ [0, 1], one has β₁(v) = β₂(v) = v for all v ∈ [0, 1].

(d) Are there equilibria at which a buyer, whose private value vi is less than 1, does not submit the bid β_i(v_i) = v_i?