Question 1

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 a simultaneous move game,neither player gets any new information as the game unfolds -- and so beliefs at the beginning of the game don't change.

Question 2

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 complete information games,there is no uncertainty about what type of player one is playing.

Question 3

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

Accepted Answer

In order for cooperation to emerge,players have to think of the long-run benefit from cooperation as outweighing the short run benefit of deviating from cooperation and then "being punished".This requires them not to discount the future too much.

Question 4

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

In the last stage,everyone simply plays the dominant strategy Nash equilibrium -- which means there is no reason not to do the same in the second to last stage and so forth.

Question 5

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

Accepted Answer

While that is possible,mixed strategy Nash equilibria can certainly have players play different pure strategies with probabilities other than 0.5.

Question 6

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

He can take 5 possible actions in the first stage.Each of these can then be combined with 5 possible actions in the second stage -- giving us 5(5)=25 possible strategies.

Question 7

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

Nash equilibrium non-credible threats are threats that,because they are taken seriously in the Nash equilibrium (that is not subgame perfect),will never have to be played.Put differently,the threat is taken seriously and alters equilibrium behavior of the other player such that the other player does not put himself in the position of having the threat executed.

Question 8

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

Accepted Answer

The strategy "never cooperate" played by everyone in the game is a subgame perfect equilibrium.

Question 9

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

The only Nash equilibrium is one where everyone plays his/her dominant strategy.

Question 10

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

Accepted Answer

A dominant strategy is best regardless of what others do -- so one would never "mix" it with another strategy and have that mix be a best response to anything done by others.

Question 11

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

Sequential games can be represented in payoff matrices with the players' strategies listed on the axes,and simultaneous games can be represented in game trees so long as player 2 has no new information when deciding on a move -- i.e.so long as player 2's nodes all lie in the same information set.

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

No,he has 32 possible strategies he can play.

Question 13

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

A strategy is a complete plan for playing the game -- which involves choosing an action at each decision node in the game.If a game is played simultaneously,each player only has one decision node -- and so the set of actions is equal to the set of possible pure strategies.

Question 14

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

Accepted Answer

Cooperation is difficult to achieve because not cooperating is a dominant strategy for each player -- i.e.regardless of whether or not the other player cooperates,it is in each player's interest not to cooperate.

Question 15

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

Accepted Answer

Subgame perfect strategies are best responses in each subgame -- regardless of whether the subgame is reached.Since they are best responses to one another,they are Nash equilibrium strategies.But Nash equilibrium allows for strategies that would play non-rationally off the equilibrium path.

Question 16

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

Accepted Answer

Both players prefer the cooperative outcome to the non-cooperative outcome -- they just can't sustain it.Thus,they would be willing to pay to be forced to cooperate.

Question 17

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 answer of Suppose a player in a sequential game...

Question 18

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

If,for a given strategy by the other player,the expected payoff of the pure strategies were not the same,then the player would best respond by placing probability 1 on the pure strategy that has the higher expected payoff.

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

A belief in such games is a probability distribution over types of players -- i.e.,a player who does not know what type of player he is playing believes the other player to be type 1 with probability p1 and type 2 with probability p2.]

Question 20

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 order to be part of a Nash equilibrium,non-credible threats have to lie "off the equilibrium path".Since one of the actions has to be chosen in the first stage of the game,nothing non-credible can therefore be threatened there -- because the first node of the game is one we reach no matter how the game is played.

Quiz 24: Strategic Thinking and Game Theory