Game Development Reference
In-Depth Information
So not only IEWDS is not order independent; in some games it is advanta-
geous not to proceed with the deletion of the weakly dominated strategies
'at full speed'. The reader may also check that the second Nash equilibrium,
( B, L ), can be found using IEWDS, as well, but not the third one, ( B, R ).
To summarise, the iterated elimination of weakly dominated strategies
can lead to a deletion of Nash equilibria,
does not need to yield a unique outcome,
can be too restrictive if we stipulate that in each round all weakly dominated
strategies are eliminated.
Finally, note that the above IEWDS Theorem 1.7 does not hold for infinite
games. Indeed, Example 1.6 applies here, as well.
1.3.3 Elimination of never best responses
Finally, we consider the process of eliminating strategies that are never best
responses to a joint strategy of the opponents. To motivate this procedure
consider the following game:
XY
A 2 , 1 0 , 0
B 0 , 1 2 , 0
C 1 , 1 1 , 2
Here no strategy is strictly or weakly dominated. However, C is a never
best response , that is, it is not a best response to any strategy of the
opponent. Indeed, A is a unique best response to X and B is a unique best
response to Y . Clearly, the above game is solved by an iterated elimination
of never best responses. So this procedure can be stronger than IESDS and
IEWDS.
Formally, we introduce the following reduction notion between the restric-
tions R and R of a given strategic game G :
N R
R
= R ,
R i
when R
i
∈{
1 ,...,n
}
R i and
R i ¬∃
i
∈{
1 ,...,n
}∀
s i
R i \
s −i
R −i s i is a best response to s −i in R.
N R when R results from R by removing from it some strategies
that are never best responses.
That is, R
 
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