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 of valuing it at $800,000, a probability of of valuing at $600,000, and a probability of 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.)

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, 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  of valuing it at $800,000, a probability of  of valuing at $600,000, and a probability of  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.)

Quizplus · Accepted Answer

In a second-bidder, sealed-bid auction (Vickrey auction), the highest bidder wins the auction but pays the price bid by the second-highest bidder. Therefore, the bank's expected revenue will be the expected second-highest bid.

Let X be the second-highest bid. Then, the expected revenue for the bank is:
E(revenue) = E(X) * P(X is the second-highest bid)

We can calculate the probability of each bidder valuing the house at each price point using the probabilities given and the assumption of independent valuations:

P(Zeke values at $800,000) = 0.11
P(Zeke values at $600,000) = 0.11
P(Zeke values at $300,000) = 0.78
P(Fanny values at $800,000) = 0.11
P(Fanny values at $600,000) = 0.78
P(Fanny values at $300,000) = 0.11
P(Heidi values at $800,000) = 0.11
P(Heidi values at $600,000) = 0.11
P(Heidi values at $300,000) = 0.78

To find the probability that X is $800,000, we need to calculate the probability that at least two bidders value the house at $800,000. There are three ways this can happen: (1) Zeke and Fanny both value at $800,000; (2) Zeke and Heidi both value at $800,000; (3) Fanny and Heidi both value at $800,000.

P(at least two bidders value at $800,000) 
= P(Zeke and Fanny value at $800,000) + P(Zeke and Heidi value at $800,000) + P(Fanny and Heidi value at $800,000)
= 0.11 * 0.11 + 0.11 * 0.11 + 0.11 * 0.11
= 0.033

To find the probability that X is $600,000, we need to calculate the probability that one bidder values at $800,000 and another bidder values at $600,000, or that two bidders value at $600,000.

P(one bidder values at $800,000 and another bidder values at $600,000) 
= 3 * (0.11 * 0.78 * 2)    (since there are three ways to choose the bidder who values at $800,000, and two ways to choose the bidder who values at $600,000)

P(two bidders value at $600,000) 
= 3 * (0.78 * 0.78 * 0.11)  (since there are three ways to choose the two bidders who value at $600,000)

P(X = $600,000) = P(one bidder values at $800,000 and another bidder values at $600,000) + P(two bidders value at $600,000)
= 3 * (0.11 * 0.78 * 2) + 3 * (0.78 * 0.78 * 0.11)
= 0.514

To find the probability that X is $300,000, we need to calculate the probability that one bidder values at $800,000, one bidder values at $600,000, and one bidder values at $300,000, or that two bidders value at $300,000.

P(one bidder values at $800,000, one bidder values at $600,000, and one bidder values at $300,000) 
= 3 * (0.11 * 0.78 * 0.78)    (since there are three ways to choose the bidder who values at $800,000)

P(two bidders value at $300,000) 
= 3 * (0.78 * 0.78 * 0.11)   (since there are three ways to choose the two bidders who value at $300,000)

P(X = $300,000) = P(one bidder values at $800,000, one bidder values at $600,000, and one bidder values at $300,000) + P(two bidders value at $300,000)
= 3 * (0.11 * 0.78 * 0.78) + 3 * (0.78 * 0.78 * 0.11)
= 0.453

Now, we can calculate the expected second-highest bid:

E(X) = $600,000 * P(X = $600,000) + $300,000 * P(X = $300,000)
= $600,000 * 0.514 + $300,000 * 0.453
= $439,020

And the bank's expected revenue is:

E(revenue) = E(X) * P(X is the second-highest bid)
= $439,020 * 1/3    (since each bidder has equal probability of being the highest bidder)
= $146,340

Therefore, the closest answer choice is D) $574,074.

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