lec3_17.pdf - Lecture 3 Strictly and Weakly Dominated...

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Lecture 3: Strictly and Weakly Dominated Strategies, More on NE 1 Strictly Dominated Strategies Def°n: Action s 0 i strictly dominates action s i (for player i ) in game ( I; ( S i ) i 2 I ; ( u i ) i 2 I ) if u i ( s 0 i ; s ° i ) > u i ( s i ; s ° i ) for all opponent strategies s ° i (i.e., s 0 i is strictly better for player i; no matter which strategy the opponent plays.) Def°n: Action s i is strictly dominated for player i in game ( I; ( S i ) i 2 I ; ( u i ) i 2 I ) if there is another action s 0 i that strictly dominates it. (Note: to be strictly dominated, s i must be a clear bad choice - there is some alternative s 0 i that would give you a strictly better payo/, against EVERY possible opponent strategy. An implication, a strictly dominated strategy cannot be a best response to any opponent strategy). 1.1 Examples 1. 0 @ A B A 2 ; 2 0 ; 0 B 0 ; 0 1 ; 1 1 A : none. 2. 0 @ A B A 2 ; 2 1 ; 1 B 0 ; 0 1 ; 1 1 A : none. (but A does WEAKLY dominate B for P1, see next section) 3. 0 @ A B A 2 ; 2 ° 1 ; 3 B 3 ; ° 1 0 ; 0 1 A ; B strictly dominates A for both players (b/c if your opponent plays A you°d prefer to play B, and if your opponent plays B you°d still prefer B; B is ALWAYS the better choice. Note, this is true even though neither player is that happy with the outcome (B,B); but it wouldn°t be possible to agree on playing (A,A), each guy would have an incentive to cheat on this agreement and deviate to B). Def°n: Action s i is weakly dominated for player i in game ( I; ( S i ) ; ( u i )) if there is another action s 0 i such that u i ( s 0 i ; s ° i ) ± u i ( s i ; s ° i ) 8
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