Arnie, Barney, and Carny happily practiced the strategy of “always bid low” for several weeks, until one day Arnie had another idea. Arnie proposed to the others: “When we all bid $L, it sometimes happens that the one who gets the week’s car values it at only $L although it is worth $H to somebody else. I’ve thought of a scheme that will increase profits for all of us.” Here is Arnie’s scheme. Every day, before Repo holds its auction, Arnie, Barney and Carny will hold a sealed-bid, second-price preauction auction among themselves in which they bid for the right to be the only high bidder in that day’s auction. The dealer who wins this preauction bidding can bid anything he likes, while the other two bidders must bid $L. A preauction auction like this is known is a “knockout.” The revenue that is collected from the “knockout” auction is divided equally among Arnie, Barney, and Carny. For this problem, assume that in the knockout auction, each bidder bids his actual value of winning the knockout auction.*
(a) If the winner of the knockout auction values the day’s used car at $H, then he knows that he can bid $H for this car in Repo’s second-price sealed-bid auction and he will get it for a price of $L. Therefore the value of winning the knockout auction to someone who values a used car at $H must be ______. The value of winning the knockout auction to someone who values a used car at $L is ______
(b) On a day when one dealer values the used car at $H and the other two value it at $L, the dealer with value $H will bid ______in the knockout auction and the other two dealers will bid ______ In this case, in the knockout auction, the dealer pays ______ for the right to be the only high bidder in Repo’s auction. In this case, the day’s used car will go to the only dealer with value $H and he pays Repo ______ for it. On this day, the dealer with the high buyer value makes a total profit of ______
(c) We continue to assume that in the knockout auction, dealers bid their actual values of winning the knockout. On days when two or more buyers value the used car at $H, the winner of the knockout auction pays ______ for the right to be the only high bidder in Repo’s auction.
(d) If Arnie’s scheme is adopted, what is the expected total profit of each of the three car dealers? (Remember to include each dealer’s share of the revenue from the knockout auction.)