First Fiddler's Bank has foreclosed on a home mortgage and is selling the house at auction.There are three bidders for the house, Ernie, Teresa, and Marilyn.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 $600,000, a probability of 1/3 of valuing at $500,000, and a probability of 1/3 of valuing it at $200,000.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)$433,333
B)$448,148
C)$600,000
D)$500,000
E)$200,000

Question

Quizplus · Accepted Answer

In a Vickrey auction, the winning bidder pays the second-highest bid. Therefore, the bank's expected revenue is the expected value of the second-highest bid. 
The probabilities of each bidder valuing the house at different prices are independent, so the probability that the second-highest bid is $600,000 is (1/3)*(1/3) = 1/9. The probability that the second-highest bid is $500,000 is (1/3)*(1/3) + (1/3)*(1/3) + (1/3)*(1/3) - (1/9) - (1/9) = 4/9. The probability that the second-highest bid is $200,000 is (1/3)*(1/3) = 1/9.
Therefore, the expected revenue for the bank is (1/9)*$600,000 + (4/9)*$500,000 + (1/9)*$200,000 = $448,148. The closest option is B.

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