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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