Game Development Reference
In-Depth Information
of strategies R 1 ,...,R n such that R i
we say that
R := ( R 1 ,...,R n ,p 1 ,...,p n )isa restriction of G . Here of course we view
each p i as a function on the subset R 1 ×
S i for i
∈{
1 ,...,n
}
S n . In what
follows, given a restriction R we denote by R i the set of strategies of player
i in R .
We now introduce the following notion of reduction between the restrictions
R and R of G :
...
×
R n of S 1 ×
...
×
S R
R
= R ,
R i
when R
i
∈{
1 ,...,n
}
R i and
R i
s i
R i s i is strictly dominated in R by s i .
i
∈{
1 ,...,n
}∀
s i
R i
\
That is, R → S R when R results from R by removing from it some strictly
dominated strategies.
In general an elimination of strictly dominated strategies is not a one step
process; it is an iterative procedure. Its use is justified by the assumption of
common knowledge of rationality.
Example 1.2
Consider the following game:
LMR
T
3 , 0
2 , 1
1 , 0
C
2 , 1
1 , 1
1 , 0
B
0 , 1
0 , 1
0 , 0
Note that B is strictly dominated by T and R is strictly dominated by M .
By eliminating these two strategies we get:
LM
T
3 , 0
2 , 1
C
2 , 1
1 , 1
Now C is strictly dominated by T , so we get:
LM
T
3 , 0
2 , 1
In this game L is strictly dominated by M , so we finally get:
M
T
2 , 1