GameTheory1 - Basic Concepts In a strategic setting, a...

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Basic Concepts • In a strategic setting, a person may not always have an choice of what is best that is independent of the actions of others • A game is an abstract model of a strategic situation • Three elements to a game – players – strategies – payoffs
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Players • Each decision maker is a player – may be individuals, firms, countries, etc. – have the ability to choose from among a set of ossible actions possible actions
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Strategies • Each course of action open to a player is a strategy – it may be a simple action or a complex plan of ction action S i is the set of strategies open to player i s i is the strategy chosen by player i , s i S i s -i is the vector of strategies chosen by all players other than player i: If there are “n” players: s -i = (s 1 ,…,s i-1 ,s i+1 ,…s n )
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Payoffs • Payoffs are measured in levels of utility obtained by the players – players are assumed to prefer higher payoffs to lower ones u 1 ( s 1 , s 2 ) denotes player 1’s payoff assuming she follows s 1 and player 2 follows s 2 u 2 ( s 2 , s 1 ) would be player 2’s payoff under the same circumstances – If many players, then u j ( s j , s -j ) is player j’s payoff if she follows s j and all other players’ strategies are s -j .
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Prisoners’ Dilemma • The Prisoners’ Dilemma is one of the most famous games studied in game theory wo suspects are arrested for a crime • Two suspects are arrested for a crime • The DA wants to extract a confession so he offers each a deal
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Prisoners’ Dilemma • The Deal – “if you fink on your companion, but your companion doesn’t fink on you, you get a one- year sentence and your companion gets a four-year sentence” – “if you both fink on each other, you will each get a three-year sentence” – “if neither finks, each will get tried for a lesser crime and each get a two-year sentence”
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Prisoners’ Dilemma • There are 4 combinations of strategies and two payoffs for each combination – useful to use a game tree or a matrix to show the payoffs • a game tree is called the extensive form • a matrix is called the normal form
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Extensive Form for the Prisoners’ Dilemma • Each node represents a decision point he dotted oval • The dotted oval means that the nodes for player 2 are in the same information set – player 2 doesn’t know player 1’s move
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Normal Form for the Prisoners’ Dilemma • Sometimes it is more convenient to represent games in a matrix
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Nash Equilibrium • Nash equilibrium involves strategic choices that, once made, provide no incentives for players to alter their ehavior behavior – best choice for each player given the other players’ equilibrium strategies
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s i is the best response for player i to rivals’ strategies s -i , denoted s i BR i ( s -i ) if u ( s , s ) u ( s’ , s ) for all s’ S i i -i i i -i i i • A Nash equilibrium is a strategy profile ( s * 1 , s * 2 ,… s * n ) such that s
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This note was uploaded on 08/24/2009 for the course MATH 262447221 taught by Professor Weisbart during the Spring '09 term at UCLA.

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GameTheory1 - Basic Concepts In a strategic setting, a...

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