gameL1 - Games in Strategic Form Definition 11.1: A...

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Games in Strategic Form De f nition 11.1: A strategic game consists of: 1. a f nite set N (the set of players), 2. for each player i N , a nonempty set A i (the set of actions available to player i ), 3. for each player i N , a preference relation % i on A = × j N A j . If the set of actions for every player is f nite, then the game is fnite . We refer to an action pro f le, a =( a j ) j N ,asan out- come .
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Note: Equivalently, we can de f ne preferences, not over outcomes, but over the consequences of those outcomes. (Sometimes it is more natural this way. In Cournot com- petition, f rms receive payo f sbasedonpro f ts rather than quantities.) g : A C If % i is the preference relation over consequences, then % i is de f ned by a % i b if and only if g ( a ) % i g ( b ) . Note: Sometimes there is randomness in determining the consequences that result from actions. We model this with a probability space, , and a function, g : A × C .T h e n a p r o f le of actions induces a lottery on C , and preferences % i must be de f nedove rth espac eo f lotteries. We can model random consequences in De f nition 11.1 by introducing nature as a player.
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Often % i can be represented by a payo f function (or utility function), u i : A R . Thenwedenotethegame by h N, ( A i ) , ( u i ) i rather than h ( A i ) , ( % i ) i .W ec a n describe f nite strategic games with two players in a table or matrix. Example: Prisoner’s Dilemma. player 2 cooperate defect player 1 cooperate 3 , 3 0 , 4 defect 4 , 0 1 , 1 Interpretations of the model: (1) The game is only played once, and players choose their actions simultaneously and independently. (2) The game or a similar game has been played in the past. Weobservethe“history ,”buttherearenostrategic links between the plays. (Maybe di f erent individuals p layedthegamep rev ious ly .) (3) By simultaneous, it is only important that each player acts in ignorance of the other players’ actions.
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Nash Equilibrium De f nition 14.1: A Nash equilibrium of a strategic game h N, ( A i ) , ( % i ) i is a pro f le of actions, a A , such that, for every player i N ,wehave ( a i ,a i ) % i ( a i i ) for all a i A i .
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This note was uploaded on 07/17/2008 for the course ECON 817 taught by Professor Peck during the Fall '07 term at Ohio State.

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gameL1 - Games in Strategic Form Definition 11.1: A...

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