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L7 - Subgame Perfect Equilibrium We recall the definition...

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Page 1 of 39 Subgame Perfect Equilibrium We recall the definition of subgame perfect equilibrium for extensive games with perfect information. D EFINITION . The subgame perfect equilibrium of an extensive game with perfect information   , , ,( ) i N H P is the strategy profile * s such that for every player i N and every nonterminal history \ h H Z for which ( ) P h i we have * * * ( , ) ( , ) h i i i h i i h h h h O s s O s s for every strategy i s of player i in the subgame ( ) h .

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Page 2 of 39 How does this concept of subgame perfect equilibrium of extensive games with perfect information extend to extensive games with imperfect information ?
Page 3 of 39 A suggestion… In t he definition, we require that each player’s strategy be optimal at every nonterminal history. So, for extensive games with imperfect information , shall we require that each player’s strategy be optimal at each of his information sets?

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Page 4 of 39 A Nash equilibrium is ( , ) L R . Really ? So L must be the best response to R , and R must be the best response to L . But why is R a good strategy at all! 3,1 0,0 L L R 0,2 1,1 L R 1 2 R M 2,2
Page 5 of 39 The Nash equilibrium ( , ) L R is not ‘subgame perfect.’ 3,1 0,0 L L R 0,2 1,1 L R 1 2 R M 2,2

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Page 6 of 39 This Nash equilibrium ( , ) M L is ‘subgame perfect.’ 3,1 0,0 L L R 0,2 1,1 L R 1 2 R M 2,2
Page 7 of 39 A more common situation: The Nash equilibrium ( , ) L R is ‘subgame perfect’ if and only if it is more probable that player 1 plays M than R . 3,1 0,2 L L R 0,2 1,1 L R 1 2 R M 2,2

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Page 8 of 39 Sequential Equilibrium A sequential equilibrium consists of a strategy profile and a belief system ( e.g. , the probability of history M is at least ½ , and that of R is at most ½ ). 3,1 0,2 L L R 0,2 1,1 L R 1 2 R M 2,2
Page 9 of 39 In the following, we restrict attention to games with perfect recall, in which every information set contains a finite number of histories.

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Page 10 of 39 Assessments Strategy Profile : 3 1 1 2 8 8 (( ( ), ( ), ( )), ( (0), (1))) L M R L R . Belief system : 1 2 3 3 { (1), (1), { , } ( ( ), ( )), ( , ) ( , )(1),( , ) ( , )(1), ( , ) ( , )(1),( , ) ( , )(1)} L L M R M R M L M L M R M R R L R L R R R R 3,1 0,2 L L R 0,2 1,1 L R 1 2 R M 2,2 This pair ( , )   is an example of assessments .
Page 11 of 39 Assessments An assessment consists of (i) a profile of behavioural strategies and (ii) a belief system consisting of a collection of probability measures, one for each information set.

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Page 12 of 39 Notations: If 1 2 3 3 { (1), (1), { , } ( ( ), ( )), ( , ) ( , )(1),( , ) ( , )(1), ( , ) ( , )(1),( , ) ( , )(1)} L L M R
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L7 - Subgame Perfect Equilibrium We recall the definition...

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