Exam 24: Strategic Thinking and Game Theory

Suppose a player in a sequential game has 2 potential decision nodes,with 5 possible actions at each node.Then he has 25 possible pure strategies.

Any non-credible threat that is part of a Nash equilibrium in a sequential game cannot be played along the Nash equilibrium path.

In mixed strategy Nash equilibria,players play each of two pure strategies with probability 0.5.

If a pooling equilibrium is played in a signaling game,the receiver will update her beliefs about the sender before settling on her best option.

Consider player n in a sequential game. a.If the player can play 3 actions from a single node,how many pure strategies does he have? b.Suppose he can play 3 actions at each of two different nodes.How many pure strategies does he have now? c.Suppose he can play 3 actions at each of three different nodes.How many pure strategies does he have now? d.Suppose he can play 3 actions at each of four different nodes.How many pure strategies does he have now? e.Suppose he can play 3 actions at each of k different nodes.How many pure strategies does he have now?

Dominant strategy Nash equilibria are efficient.

If everyone has a dominant strategy,there can be no mixed strategy equilibrium.

If we depict a simultaneous move,complete information game in a game tree,each player only has one information set no matter how many players there are in the game.

Cooperation is difficult to achieve in a Prisoners' Dilemma because each player thinks the other player might not cooperate.

Bayesian updating in a separating equilibrium implies the initially uninformed player will fully know what type he is playing when he has to make his move.

In simultaneous move Bayesian games,a player's beliefs are fully given by the probability distribution used by "Nature" to assign types.

A complete information game is a special case of an incomplete information game -- where "Nature" assigns each player a "type" with probability 1.

The Folk Theorem says that anything can happen in infinitely repeated games.

Suppose player 1 potentially moves twice in a sequential game,each time choosing from one of two possible actions -- "Left" or "Right".His first move is at the beginning of the game.He gets to move a second time if he moved "Left" the first time and after observing one of two possible actions by player 2 ("Up" or "Down").But if he moves "Right" in the first stage,he gets no further moves and the game ends after player 2 chooses one of two actions ("Up" or "Down").Draw the game tree and list all possible strategies for players 1 and 2.

If there is no pure strategy Nash equilibrium in a complete information game,there is a mixed strategy equilibrium,and if there is no mixed strategy equilibrium,there is a pure strategy equilibrium.

Consider player n in a sequential game. a.If the player can play 2 actions from a single node,how many pure strategies does he have? b.Suppose he can play 2 actions at each of two different nodes.How many pure strategies does he have now? c.Suppose he can play 2 actions at each of three different nodes.How many pure strategies does he have now? d.Suppose he can play 2 actions at each of four different nodes.How many pure strategies does he have now? e.Suppose he can play 2 actions at each of k different nodes.How many pure strategies does he have now?

In a Prisoners' Dilemma,both players are willing to pay to be forced to cooperate.

Every subgame perfect equilibrium is a Nash equilibrium but not every Nash equilibrium is a subgame perfect equilibrium.

Complete information sequential games can be represented in payoff matrices and complete information simultaneous games can be represented in game trees with information sets.

In a simultaneous move game,the number of possible pure strategies a player can play is equal to the number of actions he can choose to take.