If the government knew the costs of the two firms, it could get its hundred units at a cost.of 100 with probability 1/2, at a cost of 200 with probability 1/3, and at a cost of 300 with probability 1/6, for a total expected cost of 166.67. But, we suppose, the government doesrrt know the costs of the two firms. In addition, we suppose that all that one firm knows about the costs of its rival is the value of its own costs. That is, firm 1might know that its costs are 2, which means that it assesses probabilities 1/4 that its rivaYs costs'.are 1, 1/2 that its rival1s costs are 2, and 1/4 that its rival's costs are 3.
(a) Suppose the government seeks to find the best it can do (in terms of minimizing expected costs) in a Nash equilibrium of some mechanism. Firms must agree to participate after they learn their own costs, but then they are constrained to continue to participate. The government is not allowed to extract money from the firms; i.e., the transfers ti must be nonnegative. What is the best the government can do?
(b) Suppose we remove from (a) the constraint that transfers must he non negative. What is the best the government can do? (If you find the answer striking, see Cremer and McLean [1988].)
(c) Suppose the government seeks to find the best it can do (in terms of minimizing expected costs in a dominant strategy mechanism). What is the mimimum expected payment the government must make?