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

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Games in Strategic Form De fi nition 11.1: A strategic game consists of: 1. a fi 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 fi nite, then the game is finite . We refer to an action pro fi le, a = ( a j ) j N , as an out- come .

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Note: Equivalently, we can de fi ne preferences, not over outcomes, but over the consequences of those outcomes. (Sometimes it is more natural this way. In Cournot com- petition, fi rms receive payo ff s based on pro fi ts rather than quantities.) g : A C If % i is the preference relation over consequences, then % i is de fi 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 . Then a pro fi le of actions induces a lottery on C , and preferences % i must be de fi ned over the space of lotteries. We can model random consequences in De fi nition 11.1 by introducing nature as a player.
Often % i can be represented by a payo ff function (or utility function), u i : A R . Then we denote the game by h N, ( A i ) , ( u i ) i rather than h N, ( A i ) , ( % i ) i . We can describe fi 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. We observe the “history,” but there are no strategic links between the plays. (Maybe di ff erent individuals played the game previously.) (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 fi nition 14.1: A Nash equilibrium of a strategic game h N, ( A i ) , ( % i ) i is a pro fi le of actions, a A , such that, for every player i N , we have ( a i , a i ) % i ( a i , a i ) for all a i A i .
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gameL1 - Games in Strategic Form Definition 11.1 A...

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