4. (40 pts)

Player 1 and 2 are thinking of starting a new company. Suppose that there are two types

for Player 1: Type

-

O

(ordinary) with prob

ability

3/4 and Type

-

C

(cooperative) with prob

ability

1/4.

Player 1's type

is private information.

At the start of the game Player 1 decides whether to invest or not. If player 1 decides not to

invest, game ends.

If he decides to invest, then Player 2 decides whether to invest or not.

If Player 2 decides 2 invest, Player 1

decides to steal or split the profits.

Player 2 observes Player 1 's decision to invest or not before making her investment

decision.

Player 1 observes Player 2's decision before deciding whether to split or steal.

Payoffs from different scenarios are summ

arized in the table below:

(a)

(5 pts) Draw the game tree.

(b) (5 pts) We said that a perfect Bayes

-

Nash equilibrium has three elements. What are those three

elements?

(c) (10

pts) Show that this game does not have a separating equilibrium. Use your answer to part

b

to formulate your answer.

(d) (10 pts) Show that this game does not have a pooling equilibrium. Use your answer to part

b

to

formulate your answer.

(e) (10 pts) Let

p be the probability that Type

-

O player will invest and let q be the probability that

Player 2 will invest. Find the mixed strategy perfect Bayes

-

Nash equilibrium.