# L3 - Extensive Games Some times games are played by n...

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Page 1 of 61 Extensive Games Some times games are played by n players, in the following manner: The game starts. Player 1 i chooses and takes an action. Player 2 i chooses and takes an action. Player 3 i chooses and takes an action. The game continues until there is an outcome. Such games are called extensive games .

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Page 2 of 61 D EFINITION . An extensive game with perfect information has the following components. A set N (the set of players ). A set H of sequences (finite or infinite) that satisfies the following three properties. The empty sequence is a member of H . If 1, , () k kK aH (where K may be infinite) and LK then 1, , k kL . If an infinite sequence 1 k k a satisfies 1, , k for every positive integer L then 1 k k .
Page 3 of 61 ( H is the set of histories . A history 1, , () k kK aH is terminal if it is infinite, or if there is no 1 K a such that 1, , 1 k  . ZH is the set of terminal histories.) A function P that assigns to each nonterminal sequence (each member of \ HZ ) a member of N . ( P is the player function , Ph being the player who takes an action after the history h .) For each player iN a preference relation i on Z (the preference relation of player i on terminal histories).

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Page 4 of 61 An Extensive Game with Perfect Information: , , ,( ) i N H P .
Page 5 of 61 If H is finite, then the game is finite . If the longest history is finite, then the game has a finite horizon . with Finite Horizon with Infinite Horizon Finite Game     H H Infinite Game H H

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Page 6 of 61 is the initial history . ( ) by player ( ) ( , ) a A h Ph hH h a H a  ( ) { :( , ) } A h a h a H
Page 7 of 61   1 0 2 0 1 0 0 0 1 ( ) by (( , )) by ( ) by ( ) by player ( ) player ( ) player (( , )) player ( ) 0 1 2 ( , ) a A a a A a a aA a A h P P a P a a Ph hH h a H a a a a

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Page 8 of 61 E XAMPLE . Two people agree to use the following procedure to share two gold coins. Player 1 proposes an allocation, and Player 2 accepts or rejects. If Player 2 rejects, no one receives any gold coin. 1 2 2 2 (0,2) (2,0) (1,1) y n y n y n 2,0 0,0 1,1 0,0 0,2 0,0
Page 9 of 61 {1,2} N  { ,(2,0),(1,1),(0,2),((2,0), ),((2,0), ), ((1,1), ),((1,1), ),((0,2), ),((0,2), )} H y n y n y n  ( ) 1 P ; and ( ) 2 Ph for any nonterminal  h . 1 1 1 1 1 ((2,0), ) ((1,1), ) ((0,2), ) ((2,0), ) ((1,1), ) ((0,2), ) y y y n n n 2 2 2 2 2 ((0,2), ) ((1,1), ) ((2,0), ) ((2,0), ) ((1,1), ) ((0,2), ) y y y n n n 1 2 2 2 (0,2) (2,0) (1,1) y n y n y n 2,0 0,0 1,1 0,0 0,2 0,0

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Page 10 of 61 A Possible Strategy for Player 1 A Possible Strategy for Player 2  h : (2,0) (2,0) h : y (1,1) h : n (0,2) h : n 1 2 2 2 (0,2) (2,0) (1,1) y n y n y n 2,0 0,0 1,1 0,0 0,2 0,0
Page 11 of 61 Is this a strategy for Player 1? Is this a strategy for Player 2?  h : (2,0) (2,0) h : y (1,1) h : n (0,2) h : n 1 2 2 2 (0,2) (2,0) (1,1) y n y n y n 2,0 0,0 1,1 0,0 0,2 0,0

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Page 12 of 61 A Possible Strategy for Player 1 Is this a Strategy for Player 2?  h : (2,0) (2,0) h : y (0,2) h : n 1 2 2 2 (0,2) (2,0) (1,1) y n y n y n 2,0 0,0 1,1 0,0 0,2 0,0
Page 13 of 61 Strategies D EFINITION . A strategy of player iN in an extensive game with perfect information , , ,( ) i N H P is a function that assigns an action in () Ah to each nonterminal history \ h H Z for which P h i .

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Page 14 of 61 E XAMPLE .
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