Consider sealed-bid auctions under complete information. Bidder 1 values the object that is up for sale at $1000, while bidder 2 values it at $750. Both bidders know each others' values for the object. Independently of each others, the two bidders make a bid. The higher bidder wins the object (bidder 1 wins in case of a tie). In the first-price auction, the winner pays his/her bid. In the second-price auction, the winner pays the other's bid.

(a) Show that bidder 1 wins the object in all pure-strategy Nash equilibria of the first-price auction.

(b) Show there are pure-strategy Nash equilibria of the second-price auction where bidder 2 wins the object.