Rod’s Auction House in Bent Crankshaft, Oregon, holds sealed bid used-car auctions every Tuesday. Each used car is sold to the highest bidder at the second-highest bidder’s bid. On average, half of the cars sold at Rod’s Auction House are lemons and half are good used cars. A good used car is worth $1,000 to any buyer and a lemon is worth only $100. Buyers are allowed to look over the used cars for a few minutes before they are auctioned. Almost all of the buyers who attend the auctions can do no better than random choice at picking good cars from among the lemons. The only exception is Al Crankcase. Al can sometimes, but not always, detects a lemon by licking the oil off of the dipstick. To Al, the oil from a good car’s dipstick invariably has a sweet, lingering taste. On the other hand, the oil from the dipsticks of 1/3 of the lemons has a sour, acidic taste, while the oil from the dipsticks of the remaining 2/3 of the lemons has the same sweet taste as the oil from the good cars. Al attends every auction, licks every dipstick, and taking into account the results of his taste test, bids his expected value for every car.
(a) This auction environment is an example of a (common, private) _______value auction.
(b) Suppose that Al licks the dipsticks of 900 cars, half of which are good cars and half of which are lemons. Suppose that the dipstick oil from all of the good cars and from 2/3 of the lemons tastes sweet and the oil from 1/3 of the lemons tastes sour to Al. How many good cars will there be whose oil tastes sweet to him? _______. How many lemons will there be whose oil tastes sweet to Al? _______. How many of the 900 cars will have oil that tastes good to Al? _______. What fraction of the cars whose oil tastes sweet to Al will be good cars? _______.
(c) If Al finds that the oil on a car’s dipstick tastes sweet, what is the probability that it is a good used car?
(d) If Al finds that the oil on a car’s dipstick tastes sour, what is the probability that it is a good used car?
(e) Assuming that Al always bids his expected value for a car, given the result of his taste test, how much will Al bid for a car that tastes sweet? _______. How much will he bid for a car that tastes sour? _______.
(f) Consider a naive bidder at Rod’s Auction House, who knows that half of the cars are good and half are lemons, but has no clue at all about which ones are good. If this individual bids his expected value for a randomly selected car, how much would he bid?
(g) Given that Al bids his expected value for every used car and the naïve bidders bid the expected value of a randomly selected car, will a naïve bidder ever get a car whose oil tasted sweet to Al?
(h) What is the expected value of cars that naive bidders get if they always bid the expected value of a randomly selected car? _______. Will naive bidders make money, lose money, or break even if they follow this policy? _______.
(i) Suppose that bidders other than Al realize that they will get only the cars that taste sour to Al. If they bid the expected value of such a car, how much will they bid?
(j) Suppose that bidders other than Al believe that they will only get cars whose oil tastes sour to Al and suppose that they bid their expected value of such cars. Suppose also that for every car, Al bids his expected value, given the results of his taste test. Who will get the good cars and at what price? (Recall that cars are sold to the highest bidder at the second-highest bid.) _______.
(k) What will Al’s expected profit be on a car that passes his test? _______.