Game Theory Quiz Questions

1. Consider the following game:


1 R
(2, 2)
L M
2 2
A

(4, 1)

B

(0, 0)

A

(3, 0)

B

(0, 1)

(a) Find all the pure strategy Nash equilibria.
(b) Find all the pure strategy subgame perfect Nash equilibria. (c) Find all the pure strategy perfect Bayesian equilibria.
2. Consider the following game:


1 R
(2, 2)
L M
2 2
A B A B

(3, 0) (0, 1) (0, 1) (3, 0)




(a) Show that there is no pure strategy perfect Bayesian equilibrium for this game.
(b) Find the mixed strategy perfect Bayesian equilibrium.
3. Find all the separating and pooling equilibria of the following signaling game.
(1, 1) (2, 2)
t1

(2, 0)
2
(0, 0)

0.5
N

0.5

(0, 0)
2
(1, 0)

t2
(0, 1) (1, 1)


4. A student, player 1, has to hand in a problem set at the other end of Berkeley’s campus but needs to rush into a midterm exam. She has two options. She can deliver the problem set after the exam (call this L) and incur a late penalty. Alternatively, she can give the problem set to player 2, a random student who happens to be next to player 1 (call this S). Player 2 can either deliver the problem set on time (call this D) or throw it away in the nearest compost bin (call this T ). For player 1, the payo? is 1 if the problem set is delivered on time, ?1 if it is thrown away and 0 if it is delivered late. The payo?s for player 2 are x if he delivers and y if he throws it away.
(a) Draw the game tree for this game.
(b) What conditions do you have to place on x and y in order for player 1 to trust that player 2 will deliver the problem set in an equilibrium?




(c) Now, assume that some proportion of students are “nice guys” (N ) for which x = 1 and y = 0, while a proportion 1 ? p are “jerks” (J ) for which x = 0 and y = 1. Modify the game to allow Nature to choose what type player 2 is before the game begins. Only player 2 knows his type. Draw the new game tree.
(d) Assume p = 3 . What are the pure strategy perfect Bayesian equilibria of this game?