First Fiddler's Bank has foreclosed on a home mortgage and is selling the house at auction.There are three bidders for the house, Zeke, Fanny, and Heidi.First Fiddler's does not know the willingness to pay of these three bidders for the house, but on the basis of its previous experience, the bank believes that each of these bidders has a probability of 1/3 of valuing it at $800,000, a probability of 1/3 of valuing at $600,000, and a probability of 1/3 of valuing it at $300,000.First Fiddler's believes that these probabilities are independent among buyers.If First Fiddler's sells the house by means of a second-bidder, sealed-bid auction (Vickrey auction), what will be the bank's expected revenue from the sale? (Choose the closest option.)
A)$566,667
B)$600,000
C)$800,000
D)$574,074
E)$300,000

Question

Quizplus · Accepted Answer

In a Vickrey auction, the highest bidder wins but pays the second-highest bid. First Fiddler's expects each bidder to value the house at $800,000, $600,000, or $300,000 with equal probability. Therefore, the expected second-highest bid will be $600,000 with a probability of 1/3, $300,000 with a probability of 1/3, and $0 with a probability of 1/3. 
Expected revenue for the bank = Expected payment from the winner = expected 2nd-highest bid = (1/3)($600,000) + (1/3)($300,000) + (1/3)($0) = $200,000
But since the winner pays the second-highest bid, the bank's expected revenue = expected 2nd-highest bid = $200,000.
Therefore, the closest option is (D) $574,074.

First Fiddler's Bank Has Foreclosed on a Home Mortgage and Is