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

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