General Scooters has decided to replace its old assembly line with a new one that makes extensive use of robots. There are two contractors who would be able to build the new assembly line. General Scooters does not know exactly what it would cost either of the contractors to do this job. However its engineers and investigators have discovered that for either contractor, this cost will take one of three possible values: H, M, and L, where H > M > L. The best information that General Scooters’s investigators 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 confident that the contractors will not collude. Accountants at General Scooters suggested that General Scoooters accept sealed bids from the two contractors for constructing the assembly line and that it announces 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) In this case, as your textbook shows for the Vickrey auction, each contractor would find it in his own interest to bid his true valuation.
(a) Suppose that General Scooters uses the bidding mechanism suggested by the accountants. What is the probability that it will have to pay L? ________ (The only case where it pays L is when both contractors have a cost of L.) What is the probability that it will have to pay H to get the job done? ________(Notice that it has to pay H if at least one of the two contractors has costs of H.) What is the probability that it will have to pay M? 1/3. Write an expression in terms of the variables H, M, and L for the expected cost of the project to General Scooters. ________
(b) When the distinguished-looking, silver-haired 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 profitable for us to pay the lower of the two bids. Why 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 make?
(c) 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 contractors’ 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? (Hint: What are the contractor’s profits if he bids L and costs are L. Does he have a chance of a positive profit if he bids M?)
(d) 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? ________. Which of the two schemes will result in a lower expected cost for General Scooters, the accountants’ scheme or the chairman’s scheme?* _______.
(e) We have not yet demonstrated that the bid-padding strategies proposed 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 contractor is equally likely to have costs of H, M, or L and that he will bid 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 profits 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 M?