CEO Alice needs to choose among four projects. She hiresBob 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 theproject implemented, as summarized in the following table. The payoffs are “utility indices” whose expectation iswhat players try to maximize.Throughout the question, we assume that Alice cannot observe the types of the four projects: the four projects lookidentical to her. The timing is as follows.(1) Bob recommends a project based on his knowledge.Project type Alice’s payoff Bob’s payoffA 10 6B 6 10D -100 -1001(2) Alice chooses among three options: implementing the project recommended by Bob, implementing a randomlychosen 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 aproject, which is of types A with probability 1/4, B with probability 1/4, and D with probability 1/2. Determinewhat Alice should do in this case.(b) [15 marks] Now assume that Bob knows the types of the four projects. We model the interaction betweenAlice and Bob as a strategic form game. Bob chooses which project to recommend (A, B or D) and Alicechooses among C (accepting the recommendation), R (rejecting the recommendation and choosing randomlybetween the remaining three projects) and N (doing nothing). Write down the payoff matrix of the game andfind all Nash equilibria (in pure strategy). (Hint: first, figure out the probabilities that Alice’s randomly chosenproject 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 Aliceeven though their interests are not perfectly aligned.
Question
CEO Alice needs to choose among four projects. She hiresBob 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 theproject implemented, as summarized in the following table. The payoffs are “utility indices” whose expectation iswhat players try to maximize.Throughout the question, we assume that Alice cannot observe the types of the four projects: the four projects lookidentical to her. The timing is as follows.(1) Bob recommends a project based on his knowledge.Project type Alice’s payoff Bob’s payoffA 10 6B 6 10D -100 -1001(2) Alice chooses among three options: implementing the project recommended by Bob, implementing a randomlychosen 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 aproject, which is of types A with probability 1/4, B with probability 1/4, and D with probability 1/2. Determinewhat Alice should do in this case.(b) [15 marks] Now assume that Bob knows the types of the four projects. We model the interaction betweenAlice and Bob as a strategic form game. Bob chooses which project to recommend (A, B or D) and Alicechooses among C (accepting the recommendation), R (rejecting the recommendation and choosing randomlybetween the remaining three projects) and N (doing nothing). Write down the payoff matrix of the game andfind all Nash equilibria (in pure strategy). (Hint: first, figure out the probabilities that Alice’s randomly chosenproject 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 Aliceeven though their interests are not perfectly aligned.
Solution
(a) If Bob knows nothing more than Alice and randomly recommends a project, Alice should do nothing. This is because the expected payoff from implementing a randomly chosen project is negative. The expected payoff is calculated as follows: (1/4 * 10) + (1/4 * 6) + (1/2 * -100) = -42. Therefore, doing nothing and getting zero payoff is better than implementing a randomly chosen project.
(b) If Bob knows the types of the four projects, the payoff matrix of the game is as follows:
| A | B | D | |
|---|---|---|---|
| C | 10,6 | 6,10 | -100,-100 |
| R | E(R | A),E(R | A) |
| N | 0,0 | 0,0 | 0,0 |
Where E(R|A), E(R|B), and E(R|D) are the expected payoffs for Alice if she rejects Bob's recommendation given that he recommended A, B, or D respectively. These are calculated as follows:
E(R|A) = (1/3 * 6) + (2/3 * -100) = -64 E(R|B) = (1/3 * 10) + (2/3 * -100) = -60 E(R|D) = (1/2 * 10) + (1/2 * 6) = 8
The Nash equilibria of the game are the strategies where no player can unilaterally deviate and improve their payoff. These are (C,A), (C,B), and (N,D).
(c) Hiring Bob may help Alice even though their interests are not perfectly aligned because Bob has information that Alice does not. If Bob knows the types of the projects, he can recommend the project that maximizes Alice's payoff. Even if Bob's interests are not perfectly aligned with Alice's, it is in his best interest to recommend a project that Alice will implement, as this is the only way he can get a positive payoff. Therefore, Bob will not recommend a project of type D, and Alice can avoid the negative payoff associated with these projects.
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