4
Cooperative Games
4.5.1
For the following bimatrix games, draw the NTU and TU feasible sets.
What are the Pareto
optimal outcomes?
(a)
(0
,
4)
(3
,
2)
(4
,
0)
(2
,
3)
Solution:
The NTU feasible set is given in the figure below
(figure)
and has as Pareto optimal outcomes all vectors on the following three line segments: one
joining the points (0
,
4) and (2
,
3), one joining the points (2
,
3) and (3
,
2), and one joinng
the points (3
,
2) and (4
,
0).
The TU feasible set is given in the figure below
(figure)
and has as Pareto optimal outcomes all vectors on the line of slope 1 through the points
(2
,
3) and (3
,
2), i.e.
{
(1

t
)(2
,
3) +
t
(3
,
2) :
t
∈
R
}
=
{
(2 +
t,
3

t
) :
t
∈
R
}
.
(b)
(3
,
1)
(0
,
2)
(1
,
2)
(3
,
0)
Solution:
The NTU feasible set is given in the figure below
(figure)
and has as Pareto optimal outcomes all vectors on the line segment joining the points
(1
,
2) and (3
,
1), i.e.
{
(1

t
)(1
,
2) +
t
(3
,
1) : 0
≤
t
≤
1
}
=
{
(1 + 2
t,
2

t
) : 0
≤
t
≤
1
}
. The
TU feasible set is given in the figure below
(figure)
and has as Pareto optimal outcomes all vectors on the line of slope 1 through the point
(3
,
1), i.e.
{
(3 +
t,
1

t
) :
t
∈
R
}
.
4.5.2
Find the cooperative strategy, the TU solution, the side payment, the optimal threat strategies,
and the disagreement point for the two matrices (1) and (2) of Sections 4.1 and 4.2.
Solution:
Consider the TU game with bimatrix (1) of Section 4.1
(4
,
3)
(0
,
0)
(2
,
2)
(1
,
4)
.
The maximum total payoff is
σ
:= max
ij
(
a
ij
+
b
ij
) = 4 + 3 = 7 for the cooperative strategy
h
1
,
1
i
. Next, consider the zerosum game with matrix
A

B
=
1
0
0

3
.