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# Player 1 chooses a or b player 2 chooses c or d and

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Unformatted text preview: trictly Dominated Strategies Deﬁnition A pure strategy si is a never-best response if for all beliefs σ−i there exists σi ∈ Σi such that ui (σi , σ−i ) > ui (si , σ−i ). As shown in the preceding example, a strictly dominated strategy is a never-best response. Does the converse hold? Is a never-best response strategy strictly dominated? The following example illustrates a never-best response strategy which is not strictly dominated. 22 Game Theory: Lecture 3 Mixed Strategy Equilibrium Example Consider the following three-player game in which all of the player’s payoﬀs are the same. Player 1 chooses A or B, player 2 chooses C or D and player 3 chooses Mi for i = 1, 2, 3, 4. A B C 8 0 D 0 0 A B M1 C 4 0 D 0 4 A B M2 C 0 0 D 0 8 A B C 3 3 M3 D 3 3 M4 We ﬁrst show that playing M2 is never a best response to any mixed strategy of players 1 and 2. Let p represent the probability with which player 1 chooses A and let q represent the probability that player 2 chooses C. The payoﬀ for player 3 when she plays M2 is u3 (M2 , p , q ) = 4pq + 4(1 − p )(1 − q ) = 8pq + 4 − 4p − 4q 23 Game Theory: Lecture 3 Mixed Strategy Equilibrium Example Suppose, by contradiction, that this is a best response for some choice of p , q . This implies the following inequalities: 8pq + 4 − 4p − 4q ≥ u3 (M1 , p , q ) = 8pq ≥ u3 (M3 , p , q ) = 8(1 − p )(1 − q ) = 8 + 8pq − 8(p + q ≥ u3 (M4 , p , q ) = 3 By simplifying the top two relations, we have the following inequalities: p+q p+q ≤1 ≥1 Thus p + q = 1, and substituting into the third inequality, we have pq ≥ 3/8. Substituting again, we have p 2 − p + 3 ≤ 0 which has no 8 positive roots since the left side factors into (p − 1 )2 + ( 3 − 1 ). 2 8 4 On the other hand, by inspection, we can see that M2 is not strictly dominated. 24 Game Theory: Lecture 3 Mixed Strategy Equilibrium Rationalizable Strategies Iteratively eliminating never-best response strategies yields rationalizable strategies. ˜ Start with Si0 = Si . For each player i ∈ I and for each n ≥ 1, ˜ ˜ Sin = {si ∈ Sin−1 | ∃ σ −i ∈ ˜ ∏ Σn−1 such that j j �=i ui (si , σ−i ) ≥ ui (si� , σ−i ) ˜ for all si� ∈ Sin−1 }. ˜ ˜ Independently mix over Sin to get Σn . i ∞˜ Let Ri∞ = ∩n=1 Sin . We refer to the set Ri∞ as the set of rationalizable strategies of player i . 25 Game Theory: Lecture 3 Mixed Strategy Equilibrium Rationalizable Strategies Since the set of strictly dominated strategies is a strict subset of the set of never-best response strategies, set of rationalizable strategies represents a further reﬁnement of the set of strategies that survive iterated strict dominance. Let NEi denote the set of pure strategies of player i used with positive probability in any mixed Nash equilibrium. Then, we have NEi ⊆ Ri∞ ⊆ Di∞ , where Ri∞ is the set of rationalizable strategies of player i , and Di∞ is the set of strategies of player i that survive iterated strict dominance. 26...
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## This document was uploaded on 03/19/2014 for the course EECS 6.254 at MIT.

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