# L1 - The Prisoners Dilemma Player 2 Cooperate Defect...

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Page 1 of 53 The Prisoner’s Dilemma Player 2 Cooperate Defect Player 1 Cooperate freedom freedom freedom & award imprisonment Defect imprisonment freedom & award remission remission If you are one of them, what will YOU do?

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Page 2 of 53 Strategic Games Prisoner’s Dilemma is a typical strategic game . In general, in a strategic game: There is a finite set N of players. Player i has a set i A of available actions. An outcome is an action profile: player 1 plays a 1 , player 2 plays a 2 , player 3 plays a 3 , and so on. For player i , some outcomes are better than (‘ i ’) other outcomes. (Each player thinks differently.)
Page 3 of 53 Player 2 Cooperate Defect Player 1 Cooperate freedom freedom freedom & award imprisonment Defect imprisonment freedom & award remission remission {1,2} N . 1 { , } A C D . 2 { , } A C D . {( , ),( , ),( , ),( , )} A C C C D D C D D . ( D , C ) 1 ( C , C ) 1 ( D , D ) 1 ( C , D ). ( C , D ) 2 ( C , C ) 2 ( D , D ) 2 ( D , C ).

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Page 4 of 53 Strategic Games D EFINITION . A strategic game ,( ),( ) ii NA consists of a finite set N (the set of players ) for each player iN a nonempty set i A (the set of actions available to player i ) for each player a preference relation i on 12 n j N j A A A A A   (the preference relation of player i on the set of action profiles ).
Page 5 of 53 Payoff Functions In most cases the preference relation i can be represented by a payoff function (or utility function ) : i uA such that ( ) ( ) ii u a u b if and only if i ab . In these cases we can denote that game by ,( ),( ) N A u rather than ),( ) NA .

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Page 6 of 53 Player 2 Cooperate Defect Player 1 Cooperate 3, 3 0, 5 Defect 5, 0 1, 1 {( , ),( , ),( , ),( , )} A C C C D D C D D . 1 (( , )) 5 u D C . 1 (( , )) 3 u C C . 1 (( , )) 1 u D D . 1 (( , )) 0 u C D . ( D , C ) 1 ( C , C ) 1 ( D , D ) 1 ( C , D ). 5 3 1 0
Page 7 of 53 Nash Equilibrium * * * * * * 1 2 1 1 ( , , , , , , , ) k k k n a a a a a a A  In general, player k ’s best strategy depends on other players’ strategies. Assume * k a is the best strategy player k plays if other players do not change their strategies , that is,  * * * ( , ) ( , ) k k k k k a a a a for all kk aA . If every player thinks this way, this action profile is in equilibrium, called the Nash equilibrium .

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Page 8 of 53 Nash Equilibrium * * * * * * 1 2 1 1 ( , , , , , , , ) k k k n a a a a a a A  A Nash equilibrium is an action profile, in which every player thinks that he is playing the best strategy with respect to this profile.
Page 9 of 53 Nash Equilibrium D EFINITION . A Nash equilibrium of a strategic game ,( ),( ) ii NA is a profile * aA of actions with the property that for every player iN we have  * * * ( , ) ( , ) i i i i i a a a a for all .

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Page 10 of 53 Nash Equilibrium in other words We consider Nash equilibrium in terms of players’ best responses . Consider 1 2 1 1 ( , , , , , , ) k k k n a a a a a a Player k ’s best strategies are    { :( , ) ( , ) for all } k k k k k k k k k a A a a a a a A .
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L1 - The Prisoners Dilemma Player 2 Cooperate Defect...

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