1 Bimatrix Games – Safety Levels
1.6.1
Convert the following extensive form game to strategic form.
Solution:
(See graphical appendix at the end of this document for the tree.)
Player I has two pure strategies,
X
=
{
a,b
}
and Player II has two pure strategies as well,
Y
=
{
c,d
}
. The corresponding strategic form of this game is given by the 2
×
2 bimatrix,
±
c
d
a
1
4
(1
,
3) +
3
4
(4
,
3)
1
4
(1
,
3) +
3
4
(7
,
0)
b
1
4
(4
,
1) +
3
4
(4
,
3)
1
4
(0
,
8) +
3
4
(7
,
0)
²
=
±
c
d
a
(
13
4
,
3
) (
22
4
,
3
4
)
b
(
4
,
10
4
) (
21
4
,
2
)
²
=
±
c
d
a
(
3
1
4
,
3
) (
5
1
2
,
3
4
)
b
(
4
,
2
1
2
) (
5
1
4
,
2
)
²
1.6.2
Find the safety levels and maxmin strategies for the players in the bimatrix games,
(a)
±
(1
,
1) (5
,
0)
(0
,
5) (4
,
4)
²
Solution:
The winnings of Player I and Player II are respectively given by
A
:=
±
1 5
0 4
²
and
B
:=
±
1 0
5 4
²
.
Since