Use the following setting for questions 1-3. Consider the following static game:

Suppose this game is played for two periods.  At the beginning of period 2, the players can observe the outcome from period 1. There is no discounting.

1. First,  assume only pure-strategy  SPNE such that either player’s second period strategy DOES NOT depend on the first period outcome. How many such SPNE are in this game?

(a) 0

(b)  1

(c) 2

(d) 3

(e) 4

2. Consider only SPNE from the previous question. What is the maximum sum of the players’ payoffs that can be achieved in the two periods combined?

(a) 3

(b) 4

(c) 7

(d) 8

(e) 9

3. Now consider all possible pure-strategy SPNE such that (T,L) is played in the first period. What must be the equilibrium outcome in the second period of such SPNE?

(a)  (T,L)

(b)  (M,C)

(c)  (B,R)

(d)  (T,R)

(e) There are no such SPNE

Use the following setting for questions 4-9. Consider the following static Prisoner’s Dilemma game.

For the following parts, suppose this game is played for infinitely many times with discount factor for both players 6 ∈ [0, 1).

4. Suppose 6 = 0.  What is the maximum equilibrium payoff that player 1 can get in the first period in a SPNE of this game?

(a) -5

(b) 0

(c)  1

(d) 2

(e) 3

5. Suppose 6 = 1 for the first three rounds, and 6 = 0 for every following round.  What is the maximum equilibrium payoff that player 1 can get in the first three combined periods in a SPNE of this game?

(a) -5

(b) 0

(c)  1

(d) 2 (e) 3

6. Now we switch back to some generic value for 6 in an infinite game. Let both players adopt the following grim trigger strategy:  start with  cooperation;  cooperate  as  long  as  no  one  has ever defected before;  otherwise  defect.  What is the condition on 6 for this strategy profile to be a SPNE?

(a) 6 > 0

(b) 6 ≥ 1/3

(c) 6 ≥ 1/2

(d) 6 ≥ 3/5 (e) 6 ≥ 2/3

7. Let both players adopt the following strategy: start with cooperation; cooperate if both players cooperated  or  both players  defected in  the previous period;  defect if one player defected  and another player cooperated in the previous period. In other words, the punishment for deviating from cooperation only lasts for one period, and both players revert to cooperation immediately after. Rounded to the nearest tenth, what is the condition on 6 for this strategy profile to be a SPNE?

(a) 6 > 0