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Summary  Games and Information An Introduction to Game Theory

1.1 Definitions

What are the essential elements of a game?players, actions, payoﬀs, andinformation— PAPI

Describe natureNature is a pseudoplayer who takes random actions at speciﬁed points in the game with speciﬁed probabilities.

1.2 Dominated and dominated strategies: the prisoners dilemma

Describe a dominated strategyThe strategy sd i is a dominated strategy if it is strictly inferior to some other strategy no matter what strategies the other players choose, in the sense that whatever strategies they pick, his payoﬀ is lower with s

Describe a dominant strategyThe strategy s∗ i is a dominant strategy if it is a player’s strictly best response to any strategies the other players might pick, in the sense that whatever strategies they pick, his payoﬀ is highest with s∗

Describe the prisoners dilemmaWhenever you observe individuals in a conﬂict that hurts them all, your ﬁrst thought should be of the Prisoner’s Dilemma.

Describe a cooperative gameA cooperative game is a game in which the players can make binding commitments, as opposed to anoncooperative game, in which they cannot.

1.3 Iterated dominance: the battle of the bismarck see

Describe a weakly dominated strategyStrategy s0 i is weakly dominated if there exists some other strategy s00 i for player i which is possibly better and never worse, yielding a higher payoﬀ in some strategy proﬁle and never yielding a lower payoﬀ.

Describe an iterated dominance equilibriumAn iterated dominance equilibrium is a strategy pro ﬁle found by deleting a weakly dominated strategy from the strategy set of one of the players, recalculating to ﬁnd which remaining strategies are weakly dominated, deleting one of them, and continuing the process until only one strategy remains for each player.

Describe a zerosum gameA zerosum game is a game in which the sum of the payoﬀs of all the players is zero whatever strategies they choose. A game which is not zerosum is nonzerosum game or variable sum.

1.4 Nash equilibrium

When is there a nash equilibriumThe strategy pro ﬁle s∗ is a Nash equilibrium if no player has incentive to deviate from his strategy given that the other players do not deviate.

Describe a battle of the sexes gameThe third game we will use to illustrate Nash equilibrium is the Battle of the Sexes, a conﬂict between a man who wants to go to a prizeﬁght and a woman who wants to go to a ballet. While selﬁsh, they are deeply in love, and would, if necessary, sacriﬁce their preferences to be with each other.

Describe the coordination gamecoordination games, which share the common feature that the players need to coordinate on one of multiple Nash equilibria. Ranked Coordination has the additional feature that the equilibria can be pareto ranked.

2.1 The strategic and extensive form of a game

The strategic or normal form consists of:All possible strategy profiles
payoff function mapping 
2.3 perfect, certain, symmetric and complete information

Describe a perfect gameEach information set is a singleton

Describe a certain gameNature does not move first after any players moves

Describe a symmetric gameNo player has information different from other players when he moves, or at the end nodes

Describe a complete gameNature does not move first or her initial move is observed by every player
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