A strategy is strictly dominated if there is some strategy which strictly dominatesit. Game Theory Dr. F. Fatemi Page 100

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1 Strict Dominance Definition: For player i, a pure strategy a i strictly dominates another pure strategy a i if and only if: u i (a i, a i ) > u i (a i, a i ) a i A i In this case, a i is strictly dominated by a i. A strategy is strictly dominated if there is some strategy which strictly dominatesit. a i must yield a strictly greater payoff than a i no matter which strategy combinationis chosen by player others than i Game Theory Dr. F. Fatemi Page 100

2 We must find a single strategy (a i ) that is strictly better thana i. If a strategya i strictly dominates every other strategy in A i ; then a i is a strictlydominant strategy. L C R U 2,? 3,? 1,? M 5,? 1,? 6,? D 3,? 0,? 4,? M strictly dominates D. U and M are not dominated. Game Theory Dr. F. Fatemi Page 101

3 Iterated Elimination of Strictly Dominated Strategies Example:Quality Choice Game (One-Sided Prisoners Dilemma) Firm High Quality Low Quality Consumer Buy 2, 2-1, 3 No 0, -1 0, 0 For the firm, LQ strictly dominates HQ (3 > 2 and 0> -1). For the consumer: neither Buy nor Nois strictly dominated. Since firm is rational, it will choose LQ and will not choose HQ. The consumer knows that the firm is rational, and will not choose HQ against LQ; No is strictly better than Buy. Prediction (No, LQ) Game Theory Dr. F. Fatemi Page 102

4 When we assume rationality is common knowledge, players are able to anticipate the strategies that their opponent is not going to play for sure. 1) Eliminate strictly dominated strategies for all players. Left with reducedgame,with each player having fewer strategies 2) In the reduced game, eliminate strictly dominated strategies for all players...furtherreduced game. 3) Repeat, till no further elimination is possible. Game Theory Dr. F. Fatemi Page 103

5 Example: L C R T -1, -2 1, -3 0, -2 M -2, 1 0, 0 10, 0 B 2, 2-4, 1 3, -1 1 st step: C is strictly dominated by L Game Theory Dr. F. Fatemi Page 104

6 The reduced game: L R T -1, -2 0, -2 M -2, 1 10, 0 B 2, 2 3, -1 2 nd step: T is strictly dominated by B Game Theory Dr. F. Fatemi Page 105

7 The reduced game: L R M -2, 1 10, 0 B 2, 2 3, -1 3 rd step: R is strictly dominated by L The reduced game: L M -2, 1 B 2, 2 4 th step: M is strictly dominated by B (the prediction (B,L)) Game Theory Dr. F. Fatemi Page 106

8 Weak Dominance Definition:For player i,a pure strategy a i weakly dominates another pure strategy a i if and only if: u i (a i, a i ) u i (a i, a i ) u i (a i, a i ) > u i (a i, a i ) a i A i a i A i In this case, a i is weakly dominated by a i. A strategy is a weakly dominant strategy if it weakly dominates every other strategy. Game Theory Dr. F. Fatemi Page 107

9 Example: L C R U 2,? 3,? 1,? M 5,? 1,? 6,? D 3,? 0,? 6,? M weakly dominates B (but does not strictly dominate B) Game Theory Dr. F. Fatemi Page 108

10 Example:The money sharing game: B A Share Grab Share M/2, M/2 0, M Grab M, 0 0, 0 Then player A would at least as good as by choosing Grab regardless of the action of player B. It is called a weakly dominant strategy. In other words, for player A; Share is weakly dominated by Grab. Belot, Bhaskar, & van de Ven (2008) Can Observers Predict Trustworthiness? Investigate how good observers can predict people s behavior. Game Theory Dr. F. Fatemi Page 109

11 Iterated Elimination of Weakly Dominated Strategies The same procedure which applied for iteratively eliminating of strictly dominated actions can be applied to the weakly dominated actions. The outcome of this process may depend on the order in which the actions are eliminated. Example: L C R U 1, 1 1, 1 0, 0 D 0, 0 1, 2 1, 2 Game Theory Dr. F. Fatemi Page 110

12 Domination in Mixed Strategies: We now extend the definition of domination to the mixed strategies. Assume (A i ) denotes all possible mixed strategies for player i. Definition:For player i, a strategy α i (A i )strictly dominatesα i (A i ) if and only if U i (α i, α i ) > U i (α i, α i ) α i (A i ) In this case we say α i strictly dominates α i. Game Theory Dr. F. Fatemi Page 111

13 Example: L R T 10, 1 0, 4 M 4, 2 4, 3 B 0, 5 10, 2 M is strictly dominated by the mixed strategy where T and B each played with equal probability (½, 0, ½). Game Theory Dr. F. Fatemi Page 112