In a second-bid auction, buyers submit sealed bids, and the highest bidder obtains the item for sale but pays the seller an amount equal to the second highest bid.

a. Suppose buyers hold different private values for the item. Show that each player's dominant strategy is to bid his or her true value in this sealed bid auction. (Hint: Check that the bidder cannot gain by either understating or overstating his or her reservation price in making his or her bid.)

b. In the setting of independent, private values, the English and second- bid auctions lead to identical bids and outcomes. Explain why.

c. The major auction houses allow buyers who will not be in attendance to submit bids prior to the public auction. A representative of the house bids on the buyer's behalf up to but not in excess of the submitted bid. A submitted bid of $1,200 could well win the item for a price of $900 when the bidding stops in the actual English auction. What is the optimal submitted bid for a buyer with private value vi? Explain.