Problem 1 (This problem was inspired by news reports about the suspension of the project of building a fast rail connection between Los Angeles and San Francisco.) A consultant studies the cost of building a new project for a firm. This consultant accurately determines whether the cost of the project is low or high. The consultant then sends a report to the firm, saying either that the cost is low or that it is high. This report may or may not be truthful. After seeing the report, the firm decides to build the project or not build the project. The payo to the firm is $10 million if it builds the project and cost is low. The payo to the firm is -$20 million if it builds the project and cost is high. If the project is built, the consultant knows that it will profit from further consulting. It also knows that if its report turns out to be false, it will suer a loss of reputation. If costs are low, and the consultant reports that they are low, the consultants payo is $6 million if the firm builds the project and $1 million if it doesnt build. If costs are low and the consultant reports that they are high, the payo to the consultant is 0, whether the firm builds or not. If costs are high and the consultant reports that they are low, the payo to the consultant is $3 million if the firm builds, and 0 if it does not. If costs are high and the consultant reports that they are high, the payo to the consultant is $3 million if the firm builds and $2 million if it does not build. If the firm builds the project, its profits are $10 million if costs are low and -$20 million if it costs are high. If the firm does not build the project, the firms profits are 0, whether costs are high or low.

A) Show this game in extensive form

B) How many strategies are possible for the consultant? How many for the firm? Write down an example of a strategy for the consultant and an example of a strategy for the firm.

C) Does this game have a separating equilibrium in which the consultant reports costs truthfully? Explain.

D) Suppose that if it gets no information from the consultant, the firm believes that the probability is that the cost is low. For what values of is there a babbling equilibrium in which the firm always builds the project? For what values of is there a babbling equilibrium in which the firm never builds the project? Explain

E) Show that for the consultant, any strategy in which he reports that costs are high when they are actually low is weakly dominated.

F) Suppose that in the absence of information from the consultant, the firm believes that the probability that costs are low is . Suppose that when costs are low, the consultant always says they are low and when costs are high, the consultant randomizes and says that costs are low with probability p and high with probability 1 p. For the firm, what is the conditional probability that costs are low given that the consultant says costs are low?

G) Find a mixed strategy equilibrium in which: The consultant acts as follows: (i) when costs are low, the consultant reports that costs are low (ii) when costs are high, the consultant randomizes, and reports that they are low with probability p and high with probability (1 p). The firm acts as follows: (i) When the consultant reports high costs, the firm never builds the project (ii) when the consultant reports low costs, the firm randomizes and builds with probability q and does not build with probability 1 q. (Hint: The equilibrium probability p will depend on will depend on , but the equilibrium q will not.)