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.

Accepted Answer

The total number of possible pure strategies in a sequential game with 5 decision nodes and 2 possible actions at each node is 2^5 = 32.

Question 2

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

Accepted Answer

Every subgame perfect equilibrium (SPE) must also be a Nash equilibrium (NE) because it requires that the equilibrium strategy profile not only be optimal in the subgame being played, but also induce optimal play in any subsequent subgames that might arise. However, not every NE is an SPE because a NE does not take into account the possibility of deviations and strategies in the subsequent subgames.

Question 3

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

Accepted Answer

In simultaneous move Bayesian games, players have beliefs about other players' types, which are given by the probability distribution used by "Nature" to assign types. Therefore, a player's beliefs are fully specified by this probability distribution.

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.

Accepted Answer

This statement is true. Sequential games involve players taking turns and knowing the previous actions taken by other players, which can be represented in a payoff matrix. Simultaneous games, on the other hand, involve all players making their moves simultaneously, while not knowing the moves of other players, which 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.

Accepted Answer

In infinitely repeated Prisoners' Dilemma games, cooperation can only emerge as a subgame perfect equilibrium if players discount the future enough to make the future consequences of their actions significant enough to affect their current behavior. If players discount the future too much, then the potential long-term benefits of cooperation will not outweigh the short-term gains of defecting, and cooperation will not emerge as a subgame perfect equilibrium.

Question 6

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

Accepted Answer

In mixed strategy Nash equilibria, players can play each of their strategies with any probability distribution that makes the opponent indifferent between their strategies, not necessarily 0.5 for each strategy.

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.

Accepted Answer

In a complete information game, each player has complete and perfect information about the game, including the type of the other players. Therefore, there is no uncertainty or randomness involved in the assignment of player types. In contrast, in an incomplete information game, players do not have complete information about the game, and there may be uncertainty about the types of other players, which can affect their strategies and outcomes.

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.

Accepted Answer

In a sequential game, a pure strategy specifies an action for every decision node. With 2 nodes and 5 actions per node, the player has 5 x 5 = 25 possible pure strategies.

Question 9

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

Accepted Answer

If everyone has a dominant strategy, they will always choose the same action regardless of what the other players do. Therefore, there would be no need for any player to use a mixed strategy, and a mixed strategy equilibrium would not exist.

Question 10

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

Accepted Answer

The answer of Cooperation is difficult to achieve in a...

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.

Accepted Answer

If all players have a dominant strategy, then they will all choose that strategy regardless of what the other player(s) chooses. This means that there is only one possible outcome, which is the one that results from all players choosing their dominant strategy. Therefore, there can only be one pure strategy Nash equilibrium in this scenario.

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.

Accepted Answer

If a player has 2 possible actions and 5 possible decision nodes, then the total number of possible strategies is 2^5, which is equal to 32. Each decision node gives the player the opportunity to choose between 2 actions, and since there are 5 decision nodes, the player has 2 options at each of the 5 decision nodes. Therefore, the total number of possible strategies is 2 x 2 x 2 x 2 x 2 = 32.

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.

Accepted Answer

In a Nash equilibrium, each player's strategy is optimal given their opponent's strategies. If a player's threat is not credible, it means that they will not follow through with the threat if their opponent chooses a strategy that deviates from the Nash equilibrium. If this threat is not credible, the opponent will not believe it and will not be deterred from deviating from the Nash equilibrium. Therefore, the threat cannot be part of the Nash equilibrium path, where players are playing their optimal strategies given the opponent's strategies.

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.

Accepted Answer

If everyone has a dominant strategy in a simultaneous move game, it means that regardless of what the other players do, it is always the best option for a player to play that action. In a finitely repeated game, players have the opportunity to observe each other's actions in previous rounds and adjust their strategies accordingly. However, if there is a dominant strategy, there is no need for adjustment as it will always be the best option. Therefore, the dominant strategy will be played in all rounds of 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.

Accepted Answer

In game theory, a mixed strategy becomes a best response to another player's strategy when the expected payoffs from all the pure strategies being mixed are equal. This ensures that the player is indifferent among the options being mixed, making the mixed strategy optimal given the strategies of the other players.

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.

Accepted Answer

In a Nash equilibrium that is not subgame perfect, non-credible threats cannot be made in the first stage by the player who begins the game because the player would not have the ability to follow through on the threat in the future.

Question 17

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

Accepted Answer

In a Prisoners' Dilemma, both players would ideally want to cooperate, but due to lack of trust between them, they end up choosing the option that brings them individual benefits, which leads to a suboptimal outcome for both. Therefore, it is true that both players would willingly pay to be forced to cooperate, as it would be the best outcome for both in the long run.

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.

Accepted Answer

In a simultaneous move game, each player chooses one action from a set of possible actions simultaneously. Therefore, the number of possible pure strategies for a player is equal to the number of actions they can choose from.

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.

Accepted Answer

The answer of In a Bayesian incomplete information game, a...

Question 20

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

Accepted Answer

In an infinitely repeated Prisoners' Dilemma game, subgame perfect equilibrium can support both cooperation and defection, depending on the discount factor that reflects how players value future payoffs. If the discount factor is high enough, cooperation can be sustained as a subgame perfect equilibrium, but if it is low, defection may be the outcome in the equilibrium.

Quiz 24: Strategic Thinking and Game Theory