Find pure strategy bayes nash equilibrium of reporting game

Question

Assignment:

Problem 1

There are two workers. A worker can either be a High type or a Low type.

There is a manager who needs to make the decision of whether to pair the two workers into a team or not.

• If both workers are High types, then by working as a team each of them gets a payoff of 2.

• If both workers are Low types, then by working as a team each of them gets a payoff of 1.

• If one worker is a High type and the other is a Low type, then by working as a team each of them gets a payoff of -1. That is, a mismatch of types is counterproductive.

• If the workers are not paired up, then each of them gets a payoff of 0. The manager does not know the types of the workers, so he asks each worker to report his own type, and promises to pair the workers up if they are the same type, or not to pair them up if they are different types. Consider the game in which both workers simultaneously choose to report whether he is a High type or a Low type to the manager.

Question A. Suppose each worker knows his own type and also the type of the other worker. Is truthfully reporting one's own type always an optimal strategy regardless of what the other worker does? Explain.

Question B. Suppose each worker only knows his own type but not the type of the other worker, whereas he believes that the other worker can be a High or Low type with the same probability. Find a pure strategy Bayes Nash equilibrium of the reporting game.

Problem 2

Two players, A and B, are deciding whether to try a new restaurant or not. A has some information about the quality of the restaurant which is represented by a number a ∈ [0, 1]. B also has some information about the quality of the restaurant which is represented by another number b ∈ [0, 1]. The true quality of the restaurant is equal to a + b.

If they go to the restaurant, each of them gets a payoff of a + b. If they do not go, each of them gets a payoff of 1.

Each player knows only his own information, but not the other's. Assume that a and b are independently and uniformly distributed on [0, 1].

Suppose the players sequentially vote for whether to go or not. A votes first, and B votes after he has seen what A has voted. If both vote for going then they go, otherwise they don't. Find a sequential equilibrium of this game. (Don't forget to specify the belief system.)

Problem 3

While in his office, Bob can choose to work or shirk. If he works, his production will be high with probability p and low with probability 1 - p. If he shirks, his production will be high with probability 1 - p and low with probability p. Assume p > 0.5.

Bob's contract is written as the following: If Bob's production is high, he gets a bonus of w^H ≥ 0. If low, he gets a bonus of w^L ≥ 0.

Bob's payoff is equal to the bonus if he has shirked, or the bonus minus c > 0 if he has worked.

Question A. Will Bob work or shirk in equilibrium? Explain.

Question B. Suppose Bob has a boss, and the payoff to the boss is equal to π^H minus the bonus if the production is high, or π^L minus the bonus if the production is low, where πH, π^L are two positive constants such that π^H > π^L. What are the optimal bonus levels w^H and w^L that maximizes the payoff to the boss?

Question C. Suppose that, instead of fixing the bonuses in the contract, the boss decides how much bonus to give to Bob after he sees Bob's production level, whereas the boss does not observe whether Bob has worked or shirked. What are Bob's equilibrium bonus levels w^H∗ and w^L∗ ?