In this problem we ask how the number of bidders in a second-price, sealed-bid auction affects how much the seller can expect to receive for his object. Assume that there are two bidders who have independent, private values vi that are either 1 or 3. For each bidder, the probabilities of 1 and 3 are both 
. (If there is a tie at a bid of x for the highest bid, the winner is selected at random from among the highest bidders and the price is x.)
(a) Show that the seller’s expected revenue is 
.
(b) Now let’s suppose that there are three bidders who have independent, private values vi that are either 1 or 3. For each bidder, the probabilities of 1 and 3 are both . What is the seller’s expected revenue in this case?
(c) Briefly explain why changing the number of bidders affects the seller’s expected revenue.
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