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Unformatted text preview: libria
Example: The Penalty Kick Game.
penalty taker \ goalie
(−1, 1) (1, −1)
(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.
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
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
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- Spring '10