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the conditions of Kakutani's Theorem. The fact that for every joint mixed

strategy
m
,
best
(
m
) is non-empty is a direct consequence of the Extreme

Value Theorem stating that every real-valued continuous function on a

compact subset of

attains a maximum.

R

1.5 Iterated elimination of strategies II

The notions of dominance apply in particular to mixed extensions of finite

strategic games. But we can also consider dominance of a
pure
strategy by a

mixed
strategy. Given a finite strategic game
G
:= (
S
1
,...,S
n
,p
1
,...,p
n
),

we say that a (pure) strategy
s
i
of player
i
is
strictly dominated by
a

mixed strategy
m
i
if

∀

s
−i
∈

S
−i
p
i
(
m
i
,s
−i
)
>p
i
(
s
i
,s
−i
)
,

and that
s
i
is
weakly dominated by
a mixed strategy
m
i
if

∀

s
−i
∈

S
−i
p
i
(
m
i
,s
−i
)

≥

p
i
(
s
i
,s
−i
) and

∃

s
−i
∈

S
−i
p
i
(
m
i
,s
−i
)
>p
i
(
s
i
,s
−i
)
.

In what follows we discuss for these two forms of dominance the counter-

parts of the results presented in Section 1.3.

1.5.1 Elimination of strictly dominated strategies

Strict dominance by a mixed strategy leads to a stronger notion of strategy

elimination. For example, in the game

LR

T
2
,
1 0
,
1

M
0
,
1 2
,
1

B
0
,
1 0
,
1

the strategy
B
is strictly dominated neither by
T
nor
M
but is strictly

dominated by

2
· T
+
2
· M
.

We now focus on iterated elimination of pure strategies that are strictly

dominated by a mixed strategy. As in Section 1.3 we would like to clar-

ify whether it affects the Nash equilibria, in this case equilibria in mixed

strategies.

Instead of the lengthy wording 'the iterated elimination of strategies strict-

ly dominated by a mixed strategy' we write
IESDMS
. We have then the

following counterpart of the IESDS Theorem 1.3, where we refer to Nash

equilibria in mixed strategies. Given a restriction
G
of
G
and a joint mixed

1

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