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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.
 Spring '10
 AsuOzdaglar

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