17.5 (2) General Scooters has decided to replace its old assembly line with a new one that makes extensive use of robots. There are two con- tractors who would be able to build the new assembly line. General Scooters's industrial spies and engineers have done some exploratory re- search of their own on the costs of building the new assembly line for each NAME 217 of the two contractors. They have discovered that for each rm, this cost will take one of three possible values H, M, and L, where H > M > L. Unfortunately, General Scooters has not been able to determine whether the costs of either of the rms are H, M, or L. The best information that General Scooters's hivestigators have been able to give it is that for each contractor the probability is 1/3 that the cost is H, 1/3 that the cost is M, and 1/3 that the cost is L and that the probability distribution of costs is independent between the two contractors. Each contractor knows its own costs but thinks that the other's costs are equally likely to be H, M, or L. General Scooters is condent that the contractors will not collude. (a) Accountants at General Scooters suggested that General Scoooters accept sealed bids from the two contractors for constructing the assembly line and that it armounce that it will award the contract to the low bidder but will pay the low bidder the amount bid by the other contractor. (If there is a tie for low bidder, one of the bidders will be selected at random to get the contract.) If this is done, what bidding strategy should each of the contractors use [assuming that they cannot collude) in order to maximize their expected prots? Each would bid his true valuation. (b) Suppose that General Scooters uses the bidding mechanism suggested by the accountants. What is the probability that it will have to pay H to get the job done? 5/ 9 What is the probability that it will have to pay M ? 1/ 3 What is the probability that it will have to pay L? 1/ 9 Write an expression in terms of the variables H, M, and L for the expected cost of the project to General Scooters. 13- + M a + L1 9 9 9 (c) When the distinguishedlooking, silverhaired chairman of General Scooters was told of the accountants' suggested bidding scheme, he was outraged. \"What a stupid bidding system! Any fool can see that it is more protable for us to pay the lower of the two bids. 1Why on earth would you ever want to pay the higher bid rather than the lower one?\" he roared. A timid-looking accountant summoned up his courage and answered the chairman's question. What answer would you suggest that he mmm? The amount that contractors will bid depends on the rules of the auction. If you contract to the low bidder at the low 218 AUCTIONS [GIL 1?] bidder's bid, then all bidders will bid a higher amount than they would if you contract at the second lowest bid. {d} The chairman ignored the accountants and proposed the following plan. \"Let us award the contract by means of sealed bids, but let us do it wisely. Since we know that the contractorsI costs are either H, M, or L, we will accept only bids of H, M, or L, and we will award the contract to the low bidder at the price he himself bids. (If there is a tie, we will randomly select one of the bidders and award it to him at his bid.]\" If the chairman's scheme is adopted, would it ever be worthwhile for a contractor with costs of L to bid L? No . If he bids L, he is sure to make zero profits whether or not he gets the contract. If he bids higher than L there is a chance that he might get the contract and make a profit. {2) Suppose that the chairman's bidding scheme is adopted and that both contractors use the strategy of padding their bids in the following way. A contractor will bid M if her costs are L, and she will bid H if her costs are H or M. If contractors use this strategy, what is the expected cost of the project to General Scooters? H2 + ll"!l 1Which of the two schemes 3 3 will result in a lower expected cost for General Scooters, the acoountants' scheme or the chairman's scheme?* The accountants' scheme. (f) We have not yet demonstrated that the bid-padding strategies pro- posed above are equilibrium strategies for bidders. Here we will show that this is the case for some (but not all) values of H, M, and L. Suppose that you are one of the two contractors. You believe that the other con- tractor is equally likely to have costs of H, M, or L and that he will bid * The chairman's scheme might not have worked out so badly for Gen- eral Scooters if he had not insisted that the only acceptable bids are H, M, and L. If bidders had been allowed to bid any number between L and H, then the only equilibrium in bidding strategies would involve the use of mixed strategies, and if the contractors used these strategies, the expected cost of the project to General Scooters would be the same as it is with the second-bidder auction pr0posed by the accountants. NAME 219 H when his costs are M or H and he will bid M when his costs are L. Obviously if your costs are H, you can do no better than to bid H. If your costs are M, your expected prots will be positive if you bid H and negative or zero if you bid L or M. What if your costs are L? For what values of H, M, and L will the best strategy available to you be to bid H? 5M4H>L