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Unformatted text preview: trictly Dominated Strategies Deﬁnition
A pure strategy si is a neverbest 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
neverbest response.
Does the converse hold? Is a neverbest response strategy strictly
dominated?
The following example illustrates a neverbest response strategy which is not strictly dominated. 22 Game Theory: Lecture 3 Mixed Strategy Equilibrium Example
Consider the following threeplayer 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 neverbest 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 neverbest 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.
 Spring '10
 AsuOzdaglar

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