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How would you play this game if you were the penalty

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Unformatted text preview: libria Example: The Penalty Kick Game. penalty taker \ goalie left right left (−1, 1) (1, −1) right (1, −1) (−1, 1) No pure Nash equilibrium. How would you play this game if you were the penalty taker? Suppose you always show up left. Would this be a “good strategy”? Empirical and experimental evidence suggests that most penalty takers “randomize”→mixed strategies. 10 Game Theory: Lecture 3 Mixed Strategy Equilibrium Mixed Strategies Let Σi denote the set of probability measures over the pure strategy (action) set Si . For example, if there are two actions, Si can be thought of simply as a number between 0 and 1, designating the probability that the ﬁrst action will be played. We use σi ∈ Σi to denote the mixed strategy of player i , and σ ∈ Σ = ∏i ∈I Σi to denote a mixed strategy proﬁle. Note that this implicitly assumes that players randomize independently. We similarly deﬁne σ−i ∈ Σ−i = ∏j �=i Σj . Following von Neumann-Morgenstern expected utility theory, we extend the payoﬀ functions ui from S to Σ by ui ( σ ) = � S ui ( s ) d σ ( s ) . 11 Game Theory: Lecture 3 Mixed Strategy Equilibrium Mixed Strategy Nash Equilibrium Deﬁnition (Mixed Nash Equilibrium) A mixed strategy proﬁle σ∗ is a (mixed strategy) Nash Equilibrium if for each player i , ui (σi∗ , σ∗ i ) ≥ ui (σi , σ∗ i ) − − for all σi ∈ Σi . It is suﬃcient to check only pure strategy “deviations” when determining whether a given proﬁle is a (mixed) Nash equilibrium. Proposition A mixed strategy proﬁle σ∗ is a (mixed strategy) Nash Equilibrium if and only if for each player i , ui (σi∗ , σ∗ i ) ≥ ui (si , σ∗ i ) − − for all si ∈ Si . 12 Game Theory: Lecture 3 Mixed Strategy Equilibrium Mixed Strategy Nash Equilibria (continued) We next present a useful result for characterizing mixed Nash equilibrium. Proposition Let G = �I , (Si )i ∈I , (ui )i ∈I � be a ﬁnite strategic form game. Then, σ∗ ∈ Σ is a Nash equilibrium if and only if for each player i ∈ I , every pure strategy in the support of σi∗ is a best response to σ∗ i . − Proof idea: If a mixed strategy proﬁle is putting positive probability on a strategy that is not a best response, then shifting that probability to other strategies would improve expected utility. 13 Game Theory: Lecture 3 Mixed Strategy Equilibrium Mixed Strategy Nash Equilibria (continued) It follows that every action in the support of any player’s equilibrium mixe...
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