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
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
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
Normal (or Strategic) Form Games A matrix-based representation of games. 28
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
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
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
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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