Consider an auction with two bidders. Valuations X are uniformly and independently distributed for both bidders over the interval [0,1]. Bidder 1 knows that bidder 2 always follows the bidding strategy b2(x)=x^2.

What is the best reply of bidder 1 in a second price auction?

What is the best reply of bidder 1 in a first-price auction?

What is the expected payment of bidder 1 with valuation y in a first-price auction and in a second-price auction?

What is the ex-ante expected payment of bidder 1 in a first-price auction and in a second-price auction?