Question 1

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.

Accepted Answer

In a pooling equilibrium, all types of the sender choose the same action, so the receiver cannot update her beliefs about the sender based on the action taken.

Question 2

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?

Accepted Answer

The answer of Consider player n in a sequential game.
a.If...

Question 3

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.

Accepted Answer

In a simultaneous move, incomplete information game, player 1's strategy must account for the uncertainty about player 2's type by considering how to respond to each possible type. However, player 2, knowing his own type, will choose an action based on his type, but this does not mean player 2 "only needs to pick one action" in the strategic sense. Player 2's strategy is defined by the action chosen given his type, but from the perspective of the game's design, both players must consider their strategies across all possible information sets, including how their actions interact with the possible types or actions of the other player. The statement incorrectly simplifies player 2's strategic consideration to merely picking an action without acknowledging the strategic environment created by the game's incomplete information structure.

Question 4

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

Accepted Answer

The answer of The Folk Theorem says that anything can...

Question 5

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

Accepted Answer

In a separating equilibrium, the sender has an incentive to reveal information that distinguishes himself from others. As a result, the receiver can update his beliefs based on the sender's signal and actions during the game. Therefore, the statement is true.

Question 6

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.

Accepted Answer

In a separating equilibrium, each player chooses a different strategy based on his type, which fully reveals his type to the other player. Therefore, the initially uninformed player will fully know what type he is playing when he has to make his move, after Bayesian updating.

Question 7

A dominant strategy is one that is prevailing.

Accepted Answer

A dominant strategy is one that yields the best outcome for a player, regardless of what the other players do, not necessarily one that is prevailing or most common.

Question 8

Dominant strategy Nash equilibria are efficient.

Accepted Answer

Dominant strategy Nash equilibria are not always efficient because they can lead to outcomes that are not Pareto optimal, meaning there could be another outcome that makes at least one player better off without making any other player worse off.

Question 9

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.

Accepted Answer

In game theory, specifying a player's strategy in a sequential game includes detailing actions for every possible decision point in the game, even if the strategy involves actions intended to end the game early. This is because strategies must account for all contingencies, including how the player would react if the game does not end as anticipated.

Question 10

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

Accepted Answer

In a Prisoner's Dilemma game, not cooperating is a dominant strategy although everyone would benefit from cooperating.

Question 11

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.

Accepted Answer

In a pooling equilibrium of 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. However, on the branch that is played in equilibrium, beliefs are based on the probability distribution with which "nature" assigned types to the sender. This is a key feature of the signaling game and the basis for the information conveyed by the sender to the receiver.

Question 12

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.

Accepted Answer

The answer of Suppose player 1 potentially moves twice in...

Question 13

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)?

Accepted Answer

The answer of Consider the game depicted below.Player 1 decides...

Question 14

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.

Accepted Answer

In a simultaneous move, complete information game, all players make their move simultaneously without observing the moves of other players. Therefore, each player has only one information set, as they do not gain any additional information during the game.

Question 15

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?

Accepted Answer

The answer of Consider the following sequential move game: ...

Question 16

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.

Accepted Answer

This is a statement of the famous theorem by John Nash, which asserts that every finite game with complete information has at least one Nash equilibrium. If there is no pure strategy Nash equilibrium, it must mean that all strategies of all players are strictly dominated, which implies the existence of a mixed strategy Nash equilibrium. Conversely, if there is no mixed strategy Nash equilibrium, it must mean that every mixed strategy is also dominated by some pure strategy, which implies the existence of a pure strategy Nash equilibrium.

Question 17

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?

Accepted Answer