L6 - Extensive Games with Imperfect Information Extensive...

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Page 1 of 49 Extensive Games with Imperfect Information Extensive games with imperfect information are extensive games in which the players are imperfectly informed about some or all of the choice that have already been made.
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Page 2 of 49 E XAMPLE . 1 2 1 0,0 1,2 1,2 0,0 2,1 A B L R l r l r
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Page 3 of 49 Player 1 s information sets : { } and {( , ),( , )} L A L B . Player 2 s information set : { } L . 1 2 1 0,0 1,2 1,2 0,0 2,1 A B L R l r l r
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Page 4 of 49 D EFINITION . An extensive game 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 k K a H (where K may be infinite) and L K , then 1, , ( ) k k L a H . If an infinite sequence 1 ( ) k k a satisfies 1, , ( ) k k L a H for every positive integer L then 1 ( ) k k a H .
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Page 5 of 49 ( H is the set of histories . A history 1, , ( ) k k K a H is terminal if it is infinite, or if there is no 1 K a such that 1, , 1 ( ) k k K a H . Z H is the set of terminal histories.) A function P that assigns to each nonterminal sequence (each member of \ H Z ) a member of { } N c . ( P is the player function , ( ) P h being the player who takes an action after the history h . If ( ) P h c then chance determines the action taken after the history h .)
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Page 6 of 49 A function c f that associates with every history h for which ( ) P h c a probability measure ( ) c f h on ( ) A h , where each such probability measure is independent of every other such measure. ( ( ) c f a h is the probability that a occurs after the history h .)
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Page 7 of 49 E XAMPLE . Players information partitions : 1 information set information set { { } ,{( , ),( , )}} L A L B . 2 information set { { } } L . 1 2 1 0,0 1,2 1,2 0,0 2,1 A B L R l r l r
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Page 8 of 49 (Something new ) For each player i N a partition i of { : ( ) } h H P h i with the property that ( ) ( ) A h A h whenever h and h are in the same member of the partition. For i i I we denote by ( ) i A I the set ( ) A h and by ( ) i P I the player ( ) P h for any i h I . ( i is the information partition of player i ; a set i i I is an information set of player i .)
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Page 9 of 49 For each player i N a preference relation i on lotteries over Z (the preference relation of player i ) that can be represented as the expected value of a payoff function defined on Z .
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Page 10 of 49 Extensive Game with Imperfect Information: , , , ,( ),( ) c i i N H P f . COMPARE Extensive Game with Perfect Information: , , , ,( ) c i N H P f .
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Page 11 of 49 Player i cannot distinguish between h and h if these two histories are in the same information set: i i h I and i i h I . He only knows that some history in i I has occurred.
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