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# 14 game theory lecture 3 mixed strategy equilibrium

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Unformatted text preview: d strategy yields the same payoﬀ. Note: this characterization result extends to inﬁnite games: σ∗ ∈ Σ is a Nash equilibrium if and only if for each player i ∈ I , (i) no action in Si yields, given σ∗ i , a payoﬀ that exceeds his equilibrium − payoﬀ, (ii) the set of actions that yields, given σ∗ i , a payoﬀ less than his − equilibrium payoﬀ has σi∗ -measure zero. 14 Game Theory: Lecture 3 Mixed Strategy Equilibrium Examples Example: Matching Pennies. Player 1 \ Player 2 heads tails heads (−1, 1) (1, −1) tails (1, −1) (−1, 1) Unique mixed strategy equilibrium where both players randomize with probability 1/2 on heads. Example: Battle of the Sexes Game. Player 1 \ Player 2 ballet football (2, 1) (0, 0) ballet football (0, 0) (1, 2) This game has �wo pure Nash equilibria and a mixed Nash equilibrium t � 21 12 ( 3 , 3 ), ( 3 , 3 ) . 15 Game Theory: Lecture 3 Mixed Strategy Equilibrium Strict Dominance by a Mixed Strategy Player 1 \ Player 2 U M D Left Right (2, 0) (−1, 0) (0, 0) (0, 0) (−1, 0) (2, 0) Player 1 has no pure strategies that strictly dominate M . However, M is strictly dominated by the mixed strategy ( 1 , 0, 1 ). 2 2 Deﬁnition (Strict Domination by Mixed Strategies) An action si is strictly dominated if there exists a mixed strategy σi� ∈ Σi such that ui (σi� , s−i ) > ui (si , s−i ), for all s−i ∈ S−i . Remarks: Strictly dominated strategies are never used with positive probability in a mixed strategy Nash Equilibrium. However, as we have seen in the Second Price Auction, weakly dominated strategies can be used in a Nash Equilibrium. 16 Game Theory: Lecture 3 Mixed Strategy Equilibrium Iterative Elimination of Strictly Dominated Strategies– Revisited Let Si0 = Si and Σ0 = Σi . i For each player i ∈ I and for each n ≥ 1, we deﬁne Sin as Sin = {si ∈ Sin−1 | � σi ∈ Σn−1 such that i n ui (σi , s−i ) > ui (si , s−i ) for all s−i ∈ S−−1 }. i Independently mix over Sin to get Σn . i ∞ Let Di∞ = ∩n=1 Sin . We refer to the set Di∞ as the set of strategies of player i that survive iterated strict dominance. 17 Game Theory: Lecture 3 Mixed Strategy Equilibrium Rationalizability In the Nash equilibrium concept, each player’s action is optimal conditional on the b elief that the other players also play their Nash equilibrium strategies. The Nash Equilibrium strategy is optimal for a player given his belief about the other players strategies, and this belief is correct. We next consider a diﬀerent solution concept in which a player’s belief...
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## This document was uploaded on 03/19/2014 for the course EECS 6.254 at MIT.

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