In this problem we will ask how much a seller can expect to receive for his object in a second-price, sealed-bid auction. Assume
that all bidders have independent, private values vi which are either 0 or 1. The probability of 0 and 1 are both 1/2.
(a) Suppose there are two bidders. Then there are four possible pairs of their values (v1, v2): (0, 0), (1, 0), (0, 1), and (1, 1). Each pair of values has probability 1/4. Show that the seller’s expected revenue is 1/4. (Assume that 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.)
(b) What is the seller’s expected revenue if there are three bidders?
(c) This suggests a conjecture that as the number of bidders increases the seller’s expected revenue also increases. In the example we are considering the seller’s expected revenue actually converges to 1 as the number of bidders grows. Explain why this should occur. You do not need to write a proof; an intuitive explanation is fine.
(a) Suppose there are two bidders. Then there are four possible pairs of their values (v1, v2): (0, 0), (1, 0), (0, 1), and (1, 1). Each pair of values has probability 1/4. Show that the seller’s expected revenue is 1/4. (Assume that 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.)
(b) What is the seller’s expected revenue if there are three bidders?
(c) This suggests a conjecture that as the number of bidders increases the seller’s expected revenue also increases. In the example we are considering the seller’s expected revenue actually converges to 1 as the number of bidders grows. Explain why this should occur. You do not need to write a proof; an intuitive explanation is fine.
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