Game Development Reference
InDepth Information
1.4 Mixed extension
We now study a special case of infinite strategic games that are obtained in
a canonical way from the finite games, by allowing mixed strategies. Below
[0
,
1] stands for the real interval
.Bya
probability
distribution
over a finite nonempty set
A
we mean a function
{
r
∈
R

0
≤
r
≤
1
}
[0
,
1]
such that
a∈A
π
(
a
) = 1. We denote the set of probability distributions over
A
by Δ
A
.
Consider now a finite strategic game
G
:= (
S
1
,...,S
n
,p
1
,...,p
n
). By a
mixed strategy
of player
i
in
G
we mean a probability distribution over
S
i
.
So Δ
S
i
is the set of mixed strategies available to player
i
. In what follows,
we denote a mixed strategy of player
i
by
m
i
and a joint mixed strategy of
the players by
m
.
Given a mixed strategy
m
i
of player
i
we define
π
:
A
→
support
(
m
i
):=
{a ∈ S
i
 m
i
(
a
)
>
0
}
and call this set the
support
of
m
i
. In specific examples we write a mixed
strategy
m
i
as the sum
a∈A
m
i
(
a
)
· a
, where
A
is the support of
m
i
.
Note that in contrast to
S
i
the set Δ
S
i
is infinite. When referring to the
mixed strategies, as in the previous sections, we use the '
−i
' notation. So for
m ∈
Δ
S
1
× ...×
Δ
S
n
we have
m
−i
=(
m
j
)
j
=
i
, etc.
We can identify each strategy
s
i
∈ S
i
with the mixed strategy that puts
'all the weight' on the strategy
s
i
. In this context
s
i
will be called a
pure
strategy
. Consequently we can view
S
i
as a subset of Δ
S
i
and
S
−i
as a
subset of
×
j
=
i
Δ
S
j
.
By a
mixed extension
of (
S
1
,...,S
n
,p
1
,...,p
n
) we mean the strategic
game
(Δ
S
1
,...,
Δ
S
n
,p
1
,...,p
n
)
,
where each function
p
i
is extended in a canonical way from
S
:=
S
1
×
...
×
S
n
to
M
:= Δ
S
1
×
...
×
Δ
S
n
by first viewing each joint mixed strategy
m
=
(
m
1
,...,m
n
)
∈
M
as a probability distribution over
S
, by putting for
s
∈
S
m
(
s
):=
m
1
(
s
1
)
· ...· m
n
(
s
n
)
,
and then by putting
p
i
(
m
):=
s∈S
m
(
s
)
· p
i
(
s
)
.
The notion of a Nash equilibrium readily applies to mixed extensions. In
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