A i \u03c3 i E i \u0394 Ai satisfying the following \u03c3 i e \u0394 A e 25A mixed strategy \u03bci is

A i σ i e i δ ai satisfying the following σ i e δ

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( A i ) , σ i E i Δ ( A i ) satisfying the following: σ i ( e ) Δ ( A ( e )) . 25
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Theorem 2 (Kuhn (1953))Every mixed strategy is equivalent to the uniquebehavior strategy it generates and each behavior strategy is equivalent toevery mixed strategy that generates it. 26
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Beliefs Let ν i denote the nodes controlled by player i : ν i = { n ν ı ( n ) = i } . A belief for player i is a mapping β i E i ν i from each e E i into Δ ( ν i ) satisfying the following: β i ( e ) Δ ( e ) . (This is just requiring that beliefs respect the structure of E i , which is as- sumed to be common knowledge between the players.) Beliefs represent the players’ subjective probability of being at any given node, conditional on an information set. 27
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Normal (or Strategic) Form Games A matrix-based representation of games. 28
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Player 2 Player 1 A (4X3) Normal Form Game L C R a b c d 1, 5 3,10 -3,2 4,0 2,-1 5,3 2,2 7,3 6,5 2,-4 -4,-2 0,7 29
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Player 2 Player 1 A (2X2X3) Normal Form Game L R x y 1,5,6 3,0,10 2,-1,3 5,3,4 Player 3 plays T Player 2 Player 1 L R x y 3,5,5 13,1,0 4,5,1 8,1,3 Player 3 plays M Player 2 Player 1 L R x y 2,1,5 0,6,-2 5,2,-1 8,5,3 Player 3 plays B 30
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Player 1 Player 2 Player 2 Normal Form Representation of Extensive Form L R (a,a) (a,b) (b,a) (b,b) 1, 5 -3,2 1, 5 -3,2 2,-1 2,-1 5,3 5,3 L R a b a b Player 1 1,5 -3,2 2,-1 5,3 31
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Player 1 Player 2 Player 2 Normal Form Representation of Extensive Form L R a b 1, 5 -3,2 2,-1 5,3 L R a b a b Player 1 1,5 -3,2 2,-1 5,3 32
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