Deck 24: Strategic Thinking and Game Theory

Question 1

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

Answer 1

False

Question 2

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

Answer 2

True

Question 3

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

Answer 3

True

Question 4

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

Question 5

If players discount the future sufficiently, cooperation in infinitely repeated Prisoners' Dilemma games cannot emerge as a subgame perfect equilibrium.

Question 6

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

Question 7

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

Question 8

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.

Question 9

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

Question 10

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

Question 11

If all players in a game have a dominant strategy, then there can only be one pure strategy Nash equilibrium to the game.

Question 12

Suppose a player can play 2 possible actions and has 5 possible decision nodes in a sequential game.Then he has 10 possible strategies he can play.

Question 13

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

Question 14

If everyone has a dominant strategy in a simultaneous move game, then the action that is played by that strategy is played in all stages of any finitely repeated version of that game in any subgame perfect equilibrium.

Question 15

A mixed strategy in which positive probability is placed on more than one action can be a best response to another player's strategy only if the expected payoff from playing the pure strategies (over which the individual is mixing) is the same.

Question 16

Non-credible threats that are made in a Nash equilibrium (that is not subgame perfect) of a sequential game cannot be made in the first stage by the player who begins the game.

Question 17

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

Question 18

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.

Question 19

In a Bayesian incomplete information game, a "belief" is represented as the probability you place on your opponent playing one strategy versus another.

Question 20

In any subgame perfect equilibrium to an infinitely repeated Prisoners' Dilemma game, the players will end up cooperating.

Question 21

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.

Question 22

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?

Question 23

In a simultaneous move, incomplete information game in which player 1 is unsure of which of two types player 2 is, player 1's strategy must include an action for each possible type that player 2 might be, but player 2 only needs to pick one action since he knows what type he is.

Question 24

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

Question 25

If a separating equilibrium is played in a signaling game, the receiver will "update" his beliefs during the game.

Question 26

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.

Question 27

A dominant strategy is one that is prevailing.

Question 28

Dominant strategy Nash equilibria are efficient.

Question 29

If a player's strategy in a sequential game is to choose an action that stops the game early on, it is unnecessary to specify that player's plans for moves later on in the game when it would have been his turn to move again.

Question 30

A Prisoner's Dilemma game is one in which not cooperating is a dominant strategy despite cooperation making everyone better off.

Question 31

If a pooling equilibrium is played in a signaling game, beliefs about the sender type can take on any form along the branch of the game tree that is not played in equilibrium, but on the branch that is played, beliefs are identical to the probability distribution with which "nature" assigned types to the sender.

Question 32

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.

Question 33

Consider the game depicted below.Player 1 decides between going L or R in stage 1 and 3 of the game.Player 2 decides between going l and r in stage 2 of the game.
a.List the possible pure strategies for each player in this game and illustrate the payoffs from each pair of strategies in a matrix.
b.Is there a dominant strategy for either player?
c.Identify the subgame perfect equilibrium strategies and outcome.
d.Identify the Nash Equilibria that are not subgame perfect.
e.For each Nash Equilibrium that is not subgame perfect, explain which parts of the Nash Equilibrium strategies are non-credible.f.Suppose you have developed a drug that can be administered without the victim being aware of it.The effect of the drug is that the victim suddenly becomes gullible and believes anything he is told.You only have 1 dose of the drug and decide to auction it off to the two players right before they play each other in the game you have analyzed so far.Each player is asked to submit a sealed bid, and the highest bidder will be sold the drug at a price equal to the highest bid.In case of a tie in bids, a coin is flipped to determine who wins and pays the price that was bid.Suppose in this part that payoffs are in terms of dollars and that bids can be made in one cent increments.Suppose further that players do not consider bidding above the maximum they are willing to pay.Given that the players know each other's payoffs in the above game, what is the equilibrium price that you will be able to sell the drug for? (Hint: There are two possible answers.)
g.In part (f), we said "Suppose further that players do not consider bidding above the maximum they are willing to pay." Can you think of a Nash equilibrium to the auction that would end in a price of $8 if we had not made that statement in (f)?

Question 34

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.

Question 35

Consider the following sequential move game:
a.What are the subgame perfect equilibrium strategies in this game?
b.List all possible strategies for the two players.
c.Illustrate this game in a payoff matrix.
d.Indicate the Nash equilibria in the payoff matrix from (c).
e.What makes some of the Nash equilibria not subgame perfect?

Question 36

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.

Question 37

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?