Strategic-Form

Strategic-Form - Games in Strategic Form Econ 210C UCSB...

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Unformatted text preview: Games in Strategic Form Econ 210C UCSB March 30, 2011 Games in Strategic Form with Complete Information A game in strategic form game = { S i , u i } i N consists of a players set N = { 1, 2, . . . , n } ; a pure strategy set S i for player i N ; a payoff function u i : S = S 1 S 2 S n for player i N . Elements in S are called strategy profiles and are denoted by s = ( s 1 , s 2 , , s n ) . Timing of the game : each player i simultaneously and independently chooses a strategy s i S i ; payoff u i ( s ) is then determined and received by player i . Given s S , s- i = ( s 1 , , s i- 1 , s i + 1 , , s n ) ; s = ( s i , s- i ) . Econ 210C UCSB Paper Nash Equilibrium A strategy profile s * = ( s * 1 , s * 2 , , s * n ) S is a Nash equilibrium (NE) for = { S i , u i } i N if for all i N . u i ( s * i , s *- i ) = max s i S i u i ( s i , s *- i ) . Non-existence of a NE in pure strategies: N = { 1, 2 } ; S i = { H , T } ; u 1 ( s ) = 1 if s 1 = s 2 ; u 1 ( s ) =- 1 is s 1 6 = s 2 ; u 1 ( s ) + u 2 ( s ) = for all s S . Heads Tails Heads 1 -1 -1 1 Tails -1 1 1 -1 Matching Pennies Econ 210C UCSB Paper Given a strategic form game = { S i , u i } in N , X i = { x i : S i < + | s i S i x i ( s i ) = 1 } ; X = X 1 X 2 X n ; for all x = ( x 1 , x 2 , , x n ) X , u i ( x ) = s S u i ( s ) x 1 ( s 1 ) x 2 ( s 2 ) x n ( s n ) = s i S i u i ( s i , x- i ) x i ( s i ) (1) where u i ( s i , x- i ) = s- i S- i u i ( s i , s- i ) x- i ( x- i ) , x- i ( s- i ) = x 1 ( s 1 ) x i- 1 ( s i- 1 ) x i + 1 ( s i + 1 ) x n ( s n ) . A mixed strategy profile x * X is a NE if u i ( x * i , x *- i ) = max x i X i u i ( x i , x *- i ) . (2) Econ 210C UCSB Paper A strategy s i S i is strictly dominated by a strategy x i X i if for all s- i S- i , u i ( x i , s- i ) > u i ( s i , s- i ) . Any mixed strategy assigning positive probability to a strictly dominated pure strategy is strictly dominated. A pure strategy s i S i is strictly dominant if any s i S i with s i 6 = s i is strictly dominated by s i . Econ 210C UCSB Paper A strategy s i S i is weakly dominated by x i X i if u i ( x i , s- i ) u i ( s i , s- i ) for all s- i S- i and u i ( x i , s- i ) > u i ( s i , s- i ) for at least one s- i S- i . A strategy s i S i is weakly dominant if any s i S i with s i 6 = s i is weakly dominated by s i . Econ 210C UCSB Paper A Special Property of Nash Equilibrium Given x i X i , supp ( x i ) = { s i S i | x i ( s i ) > } . Lemma Let x * B be a Nash equilibrium. Then, for all i, u i ( s i , x *- i ) = u i ( x * ) , s i supp ( x * i ) . (3) Proof....
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Strategic-Form - Games in Strategic Form Econ 210C UCSB...

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