MIXED STRATEGIES
GAETANO BLOISE
Department of Economics, University of Rome III
February 28, 2011 - June 6, 2011
Mixed extension
The mixed extension of the (finite) game in normal form G = (Si , ui )i∈J , is given
by the game in normal form ∆ (G) = (Σi , ui )i∈J , where, for every player i in J, the
strategy space is Σi is the set of all probability measures σi of the original strategy
space Si and, with some abuse of notation, ui : Σ → R represents the expected
utility, that is,
X
ui (σ) =
ui (s) σ (s) .
s∈S
In particular, given a mixed strategy σi in Σi , σi (si ) is the probability of strategy
si in Si and, given a strategy profile σ in Σ, σ (s) is the probability of strategy
profile s in S, that is,
Y
σ (s) =
σi (si ) = σi (si ) σ−i (s−i ) .
i∈J
Notice that a pure strategy si in Si can be identified with the mixed strategy σi in
Σi with σi (si ) = 1, so that Si ⊂ Σi . Finally, expected utility decomposes as
X
ui (σi , σ−i ) =
ui (si , σ−i ) σi (si ) .
si ∈Si
Optimal response
Define
βi (σ−i ) = {σi ∈ Σi : ui (σi , σ−i ) ≥ ui (σi∗ , σ−i ) for every σi∗ ∈ Σi } .
and
bi (σ−i ) = {si ∈ Si : ui (si , σ−i ) ≥ ui (s∗i , σ−i ) for every s∗i ∈ Si } .
The map bi : Σ−i → Si gives optimal pure strategies, whereas the map βi : Σ−i →
Σi gives optimal mixed strategies, conditional on a given mixed strategy profile σ−i
in Σ−i .
Lemma. Given any σ−i in Σ−i ,
βi (σ−i ) = {σi ∈ Σi : σi (si ) > 0 only if si ∈ bi (σ−i )} .
In other terms, a mixed strategy is optimal if and only if it assigns strictly positive
probability only to optimal pure strategies.
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Proof. Given any mixed strategy σi in Σi ,
X
ui (s∗i , σ−i ) σi (s∗i )
ui (σi , σ−i ) =
s∗
i ∈Si
≤
X s∗
i ∈Si
=
=
max ui (si , σ−i ) σi (s∗i )
si ∈Si
X
σi (s∗i )
max ui (si , σ−i )
si ∈Si
s∗
i ∈Si
max ui (si , σ−i ) .
si ∈Si
It follows that
ui (σ−i ) = max ui (σi , σ−i ) = max ui (si , σ−i ) .
σi ∈Σi
si ∈Si
To prove the claim, observe that a mixed strategy σi lies in βi (σ−i ) if and only if
ui (σ−i ) = ui (σ−i , σ−i )
or, equivalently,
X
(*)
(ui (σ−i ) − ui (si , σ−i )) σi (si ) = 0.
si ∈Si
Restriction (*) requires that, for every pure strategy si in Si ,
(ui (σ−i ) − ui (si , σ−i )) σi (si ) = 0,
as ui (σ−i ) − ui (si , σ−i ) ≥ 0 e σi (si ) ≥ 0. Therefore, a mixed strategy σi lies in
βi (σ−i ) if and only if, for every pure strategy si in Si ,
σi (si ) > 0 only if ui (σ−i ) = ui (si , σ−i )
o, equivalentemente,
ui (σ−i ) > ui (si , σ−i ) only if σi (si ) = 0.
Observing that, for every pure strategy si in Si ,
si ∈ bi (σ−i ) if and only if ui (σ−i ) = ui (si , σ−i ) ,
this proves the claim.
Existence (Nash)
The Existence Theorem requires a fundamental theorem of analysis.
Fixed-Point Theorem. Given a non-empty, convex, compact set S is some Euclidean space, any continuous map f : S → S admits a fixed point, that is, there
exists s∗ in S such that f (s∗ ) = s∗ .
Esistence Theorem (Nash). Any mixed extension ∆ (G) of a finite game in
normal form G admits a Nash equilibrium, that is, there exists a strategy profile σ ∗
in Σ such that, for every player i in J,
∗
∗
ui σi∗ , σ−i
≥ ui σi , σ−i
, for every σi ∈ Σi .
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Proof. Consider the continuous map f : Σ → Σ defined by
+
f (σ)i (s∗i ) =
σi (s∗i ) + (ui (s∗i , σ−i ) − ui (σi , σ−i ))
P
+,
1 + si ∈Si (ui (si , σ−i ) − ui (σi , σ−i ))
where, for every x in R, x+ = max {x, 0}. By the Fixed-Point Theorem, there exists
a strategy profile σ ∗ in Σ such that σ ∗ = f (σ ∗ ). Notice that, at this fixed point,
X
+
+
∗
∗
∗
∗
(*) σi∗ (s∗i )
ui si , σ−i
− ui σi∗ , σ−i
= ui s∗i , σ−i
− ui σi∗ , σ−i
.
si ∈Si
Suppose that
X
(**)
+
∗
∗
ui si , σ−i
− ui σi∗ , σ−i
> 0.
si ∈Si
By condition (*), it immediately follows that
+
∗
∗
ui s∗i , σ−i
− ui σi∗ , σ−i
= 0 if and only if σi∗ (s∗i ) = 0.
Thus,
+ ∗ ∗
∗ ∗
∗
∗
∗
∗
− ui σi∗ , σ−i
σi (si ) = ui s∗i , σ−i
− ui σi∗ , σ−i
σi (si ) .
ui s∗i , σ−i
This implies that
0
=
X
∗ ∗
∗
∗
ui s∗i , σ−i
− ui σi∗ , σ−i
σi (si )
s∗
i ∈Si
=
X
+ ∗ ∗
∗
∗
ui s∗i , σ−i
− ui σi∗ , σ−i
σi (si )
s∗
i ∈Si
>
0,
a contradiction.
As restriction (**) cannot be fulfilled, for every s∗i in Si ,
∗
∗
ui s∗i , σ−i
− ui σi∗ , σ−i
≤ 0.
By the characterization of optimal response in mixed strategies, this suffices to
prove that strategy profile σ ∗ in Σ is a Nash equilibrium of game ∆ (G).
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