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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.
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
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
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
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