(Aghion and Tirole's model of delegation) CEO Alice needs to choose among four projects. She hires Bob to recommend a project to her. The four projects are of three types, A, B, and D. One project is of type A, one project is of type B and two projects are of type D. Both Alice's and Bob's payoffs depend on the types of the project implemented, as summarized in the following table. The payoffs are "utility indices" whose expectation is what players try to maximize. Throughout the question, we assume that Alice cannot observe the types of the four projects: the four projects look identical to her. The timing is as follows. (1) Bob recommends a project based on his knowledge. Project type Alice's payoff Bob's payoff 10 6 -100 A B D 6 10 -100 1 (2) Alice chooses among three options: implementing the project recommended by Bob, implementing a randomly chosen project not recommended by Bob, and doing nothing. Doing nothing gives both players zero payoff. (a) [5 marks] First, assume that Bob knows nothing more than Alice. Therefore, he randomly recommends a project, which is of types A with probability 1/4, B with probability 1/4, and D with probability 1/2. Determine what Alice should do in this case. (b) [15 marks] Now assume that Bob knows the types of the four projects. We model the interaction between Alice and Bob as a strategic form game. Bob chooses which project to recommend (A, B or D) and Alice chooses among C (accepting the recommendation), R (rejecting the recommendation and choosing randomly between the remaining three projects) and N (doing nothing). Write down the payoff matrix of the game and find all Nash equilibria (in pure strategy). (Hint: first, figure out the probabilities that Alice's randomly chosen project is of type A, B and D if she rejects Bob's recommendation.) (c) [5 marks] Discuss why hiring Bob (whose salary is negligible compared with project payoffs) may help Alice even though their interests are not perfectly aligned. (Limited to 50 words.)