Every day the Repo finance company holds a sealed-bid, second-price auction in which it sells a repossessed automobile. There are only three bidders who bid on these cars, Arnie, Barney, and Carny. Each of these bidders is a used-car dealer whose willingness to pay for another used car fluctuates randomly from day to day in response to the variation in demand at his car lot. The value of one of these used cars to any dealer, on any given day is a random variable which takes a high value $H with probability 1/2 and a low value $L with probability 1/2. The value that each dealer places on a car on a given day is independent of the values placed by the other dealers. Each day the used-car dealers submit written bids for the used car being auctioned. The Repo finance company will sell the car to the dealer with the highest bid at the price bid by the second-highest bidder. If there is a tie for the highest bid, then the second-highest bid is equal to the highest bid and so that day’s car will be sold to a randomly selected top bidder at the price bid by all top bidders.
(a) How much should a dealer bid for a used car on a day when he places a value of $H on a used car? ________. How much should a dealer bid for a used car on a day when he places a value of $L on a used car? ________.
(b) If the dealers do not collude, how much will Repo get for a used car on days when two or three dealers value the car at $H? ________. How much will Repo get for a used car on days when fewer than two dealers value the car at $H? ________.
(c) On any given day, what is the probability that Repo receives $H for that day’s used car? ________. What is the probability that Repo receives $L for that day’s used car? 1/2. What is Repo’s expected revenue from the sale? ________.
(d) If there is no collusion and every dealer bids his actual valuation for every used car, what is the probability on any given day that Arnie gets a car for a lower price than the value he places on it? (This will happen only if the car is worth $H to Arnie and $L to the other dealers.) ________. Suppose that we measure a car dealer’s profit by the difference between what a car is worth to him and what he pays for it. On a randomly selected day, what is Arnie’s expected profit? ________.
(e) The expected total profit of all participants in the market is the sum of the expected profits of the three car dealers and the expected revenue realized by Repo. Used cars are sold by a sealed-bid, second-price auction and the dealers do not collude. What is the sum of the expected profits of all participants in the market?