Problem Set 5

Which is the pure strategy Nash equilibrium of this stage game (if it is played only once)?

Suppose that the game is repeated for two periods. What is the outcome from the subgame perfect Nash equilibrium of the whole game:

Suppose that there is a probability $p$ that the game continues next period and a probability $(1-p)$ that it ends. What is the threshold p^*p∗ such that when p \geq p^*p≥p∗ (Middle, Middle) is sustainable as a subgame perfect equilibrium by grim trigger strategies, but when p < p^*p<p∗ playing Middle in all periods is not a best response? [Here the grim strategy is: play Middle if the play in all previous periods was (Middle, Middle); play Right otherwise.]

Which are the pure strategy Nash equilibria of this stage game? There can be more than one.

Suppose that the game is repeated for two periods. Which of the following outcomes could occur in some subgame perfect equilibrium? (There might be more than one).

There is a probability $p$ that the game continues next period and a probability $(1-p)$ that it ends. The game is repeated indefinitely. Which statement is true? [Grim trigger in (c) and (d) is player 1 playing Not play and player 2 playing Distrust forever after a deviation from ((Play,Share), (Trust)).]

• There are two “statuses” that player i might be in during any period: “normal” and “revenge”;
• In a normal status player i cooperates;
• In a revenge status player i defects;
• From a normal status, player i switches to the revenge status in the next period only if the other player defects in this period;
• From a revenge status player i automatically switches back to the normal status in the next period regardless of the other player’s action in this period.

Consider an infinitely repeated game so that with probability $p$ that the game continues to the next period and with probability $(1-p)$ it ends.

True or False:

When player 1 uses the above-described “tit for tat” strategy and starts the first period in a revenge status (thus plays defect for sure), in any infinite payoff maximizing strategy, player 2 plays defect in the first period

• There are two “statuses” that player i might be in during any period: “normal” and “revenge”;
• In a normal status player i cooperates;
• In a revenge status player i defects;
• From a normal status, player i switches to the revenge status in the next period only if the other player defects in this period;
• From a revenge status player i automatically switches back to the normal status in the next period regardless of the other player’s action in this period.

Consider an infinitely repeated game so that with probability $p$ that the game continues to the next period and with probability $(1-p)$ it ends.

What is the payoff for player 2 from always cooperating when player 1 uses this tit for tat strategy and begins in a normal status? How about always defecting when 1 begins in a normal status?

• There are two “statuses” that player i might be in during any period: “normal” and “revenge”;
• In a normal status player i cooperates;
• In a revenge status player i defects;
• From a normal status, player i switches to the revenge status in the next period only if the other player defects in this period;
• From a revenge status player i automatically switches back to the normal status in the next period regardless of the other player’s action in this period.

Consider an infinitely repeated game so that with probability $p$ that the game continues to the next period and with probability $(1-p)$ it ends.

What is the threshold p^*p∗ such that when p\geq p^*p≥p∗ always cooperating by player 2 is a best response to player 1 playing tit for tat and starting in a normal status, but when p<p^*p<p∗ always cooperating is not a best response?