Lecture 3
Source:
/1
/
D.R.Anderson, D.J.Sweeney, T.A.Williams. Quantitative Methods for Business, South
Western College Publishing, 11th edition.
Chapters 4, 5.
Contents:
3.1
Decision making with probabilities
3.2
Utility and decision making
3.3
Utility: other considerations
3.1
Decision making with probabilities
In many decisionmaking situations, we can obtain probability assessments for the states of
nature. When such probabilities are available, we can use the expected value approach to
identify the best decision alternative. Let us first define the expected value of a decision al
ternative and then apply it to the PDC problem.
Let
N
= the number of states of nature ,
P
(
s
j
) = the probability of state of nature
s
j
.
Because one and only one of the
N
states of nature can occur, the probabilities must satisfy two
conditions:
P
(
s
j
)
> 0
for all states of nature
(3.1)
1
)
(
)
(
)
(
)
(
2
1
1
=
+
⋅
⋅
⋅
+
+
=
∑
=
N
N
j
j
s
P
s
P
s
P
s
P
(3.2)
The
expected value (EV)
of decision alternative
d
i
is defined as follows:
)
(
)
(
)
(
1
,
j
N
j
i
j
i
s
d
V
s
P
d
EV
∑
=
=
(3.3)
Let us return to the PDC problem to see how the expected value approach can be applied.
Example (3.1) of the expected value
approach in PDC problem (Lecture 2)
Let the a priori probability for
s
1
(strong demand) be 0.8 and for
s
2
(week demand) –
0.2, i.e.,
8
.
0
)
(
1
=
s
P
and
2
.
0
)
(
2
=
s
P
.
Using the payoff values in
Table 2.1
and equation
(3.3)
, we compute the expected value for
each of the three decision alternatives as follows:
8
.
7
)
7
(
2
.
0
)
8
(
8
.
0
)
(
1
=
+
=
d
EV
2
.
12
)
5
(
2
.
0
)
14
(
8
.
0
)
(
2
=
+
=
d
EV
2
.
14
)
9
(
2
.
0
)
20
(
8
.
0
)
(
3
=

+
=
d
EV
Thus, using the expected value approach, we find that the large condominium complex, with
an expected value of
$14.2
million, is the recommended decision.
The calculations required to identify the decision alternative with the best expected value can
be conveniently carried out on a decision tree.
Figure 3.1
shows the decision tree for the
PDC problem with stateofnature branch probabilities. Working backward through the
decision tree, we first compute the expected value at each chance node. That is, at each
chance node, we weight each possible payoff by its probability of occurrence. By doing so,
we obtain the expected values for nodes
2, 3,
and
4,
Because the decision maker controls the branch leaving decision node
1
.and because we are
trying to maximize the expected profit, the best decision alternative at node
1
is
d
3
.
Thus, the
decision tree analysis leads to a recommendation of
d
3
with an expected value of
$14.2