Final Exam

Find the strictly dominant strategies (click all that apply: there may be zero, one or more and remember the difference between strictly dominant and strictly dominated):

Find the weakly dominated strategies (click all that apply: there may be zero, one or more):

Which strategies survive the process of iterative removal of strictly dominated strategies (click all that apply: there may be zero, one or more)?

Find all strategy profiles that form pure strategy Nash equilibria (click all that apply: there may be zero, one or more):

Which of the following strategies form a mixed strategy Nash equilibrium? ($p$ corresponds to the probability of 1 playing $\text{b}$ and $1-p$ to the probability of playing $\text{c}$; $q$ corresponds to the probability of 2 playing $y$ and $1-q$ to the probability of playing $z$).

• One island is occupied by Army 2, and there is a bridge connecting the island to the mainland through which Army 2 could retreat.
• Stage 1: Army 2 could choose to burn the bridge or not in the very beginning.
• Stage 2: Army 1 then could choose to attack the island or not.
• Stage 3: Army 2 could then choose to fight or retreat if the bridge was not burned, and has to fight if the bridge was burned.

First, consider the blue subgame. What is a subgame perfect equilibrium of the

blue subgame?

• One island is occupied by Army 2, and there is a bridge connecting the island to the mainland through which Army 2 could retreat.
• Stage 1: Army 2 could choose to burn the bridge or not in the very beginning.
• Stage 2: Army 1 then could choose to attack the island or not.
• Stage 3: Army 2 could then choose to fight or retreat if the bridge was not burned, and has to fight if the bridge was burned.

What is the outcome of a subgame perfect equilibrium of the whole game?

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∗ ((Play,Share), (Trust)) is sustainable as a subgame perfect equilibrium by a grim trigger strategy, but when p<p^*p<p∗ ((Play,Share), (Trust)) can’t be sustained as a subgame perfect equilibrium?

[Here a trigger strategy is: player 1 playing Not play and player 2 playing Distrust forever after a deviation from ((Play,Share), (Trust)).]

• There are two players.
• The payoffs to player 2 depend on whether 2 is a friendly player (with probability p) or a foe (with probability 1−p).
• Player 2 knows if he/she is a friend or a foe, but player 1 doesn’t know.

See the following payoff matrices for details.

with probability $p$

with probability $1-p$

When $p=1/4$, which is a pure strategy Bayesian equilibrium:

(1’s strategy; 2’s type – 2’s strategy)

• If 1 stays out, the payoff to both players is (0, 3).

Player 2 is already in the market and chooses (simultaneously) whether to fight

player 1 if there is entry

• The payoffs to player 2 depend on whether 2 is a normal player (with prob $1-p$) or an aggressive player (with prob $p$).

See the following payoff matrices for details.

with probability $p$

with probability $1-p$

Player 2 knows if he/she is normal or aggressive, and player 1 doesn’t know. Which are true (click all that apply, there may be zero, one or more):