Game Development Reference
In-Depth 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 non-empty 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
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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