Suppose we modify problem 6 in two ways. First suppose that the firms have costs of 1, 2, or 3, with each level of cost equally likely, and with the cost levels of the two firms independent of one another. (Each firm knows its own cost but not the costs of its rival.) Suppose the government solicits bids from the firms. If they bid the same amount, the order is spliti if one firm bids less, it is given the entire order. And (the second change) a firm is paid not what it bid but what the other firm bid.

(a) If we construct from this mechanism the corresponding game of incomplete information, does this game have a pure strategy Nash equilibrium? Does it have more than one? If there is more than one, is there any reason to think that one equilibrium is more tenable than the others?

(b) For the equilibria you find in (a) construct the corresponding direct revelation games.
(This sort of auction, where the winning bidder is rewarded according to the bid made by the "second" best bid, is called a Vickrey auction. Almost any treatment of auction theory will tell you how to analyze this sort of auction, if you cant figure it out for yourself.)

Problem 6

Consider the toy problem of section 18.2. Imagine that the government institutes the following procedure: The two firms are asked to bid (simultaneously and independently) on the prices they will charge. They may not name a price greater than 4 or less than zero. (Of course, each knows its own unit cost when it does this.) If they name the same price, the order is split between them. If they name a different price, the order is given entirely to the firm naming the lower price. In both cases, the firm is paid per unit what it bids.

(a) If we construct from this mechanism the corresponding game (between the firms) of incomplete information, does the game have a pure strategy Nash equilibrium?

(b) If not, can you find a symmetric mixed strategy equilibrium?

(c) If you find any equilibrium, what is the corresponding direct revelation mechanism?