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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 Γ = { 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 Γ = { 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 ) . Nonexistence 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 Γ = { 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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