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 it’s report turns out to be false, it will su↵er a loss of reputation.

If costs are low, and the consultant reports that they are low, the consultant’s payo↵ is $6 million if the firm builds the project and $1 million if it doesn’t 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 firm’s profits are 0, whether costs are high or low.

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.