First Fiddler's Bank has foreclosed on a home mortgage and is selling the house at auction. There are three bidders for the house, Jesse, Shelia, and Elsie. First Fiddler's does not know the willingness to pay of any of these bidders but on the basis of its previous experience believes that each of them has a probability of of valuing the house at $800,000, a probability of of valuing it at $500,000, and a probability of of valuing it at $300,000. First Fiddler's believes that these probabilities are independent between 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 answer.)

Question

First Fiddler's Bank has foreclosed on a home mortgage and is selling the house at auction. There are three bidders for the house, Jesse, Shelia, and Elsie. First Fiddler's does not know the willingness to pay of any of these bidders but on the basis of its previous experience believes that each of them has a probability of  of valuing the house at $800,000, a probability of  of valuing it at $500,000, and a probability of  of valuing it at $300,000. First Fiddler's believes that these probabilities are independent between 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 answer.)

Quizplus · Accepted Answer

Under the Vickrey auction, the highest bidder wins the auction but pays the price of the second-highest bidder. To find the bank's expected revenue, we need to first determine the expected values of each bid.

For each bidder, the expected value of their bid can be calculated as follows:

E(800,000) = 0.11(800,000) + 0.89(0) = 88,000
E(500,000) = 0.11(500,000) + 0.89(0) = 55,000
E(300,000) = 0.11(300,000) + 0.89(0) = 33,000

The expected value of the second-highest bid can be calculated as follows:

E(second-highest bid) = Σ (probability of bidder valuing the house at a certain price * expected value of their bid)
= (0.11 * 800,000) + (0.11 * 500,000) + (0.11 * 300,000)
= 123,000

Therefore, the bank's expected revenue would be the expected value of the second-highest bid (i.e., $123,000) subtracted from the highest bid. The highest bid can be determined by finding the bidder with the highest expected value of their bid:

Jesse: 88,000
Shelia: 55,000
Elsie: 33,000

Thus, Jesse has the highest expected value of their bid and is the highest bidder. Therefore, the bank's expected revenue would be:

Expected revenue = highest bid - E(second-highest bid)
= 800,000 - 123,000
= 677,000

The closest answer choice is B) $533,333.33.

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