# Decision Theory

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```					                 Chapter 14
Decision Analysis

Problem Formulation
Decision Making without Probabilities
Decision Making with Probabilities
Risk Analysis and Sensitivity Analysis
Decision Analysis with Sample Information
Computing Branch Probabilities
Utility and Decision Making

Dr. C. Lightner           1
Fayetteville State University
Problem Formulation

A decision problem is characterized by decision alternatives,
states of nature, and resulting payoffs.
The decision alternatives are the different possible strategies the
decision maker can employ.
The states of nature refer to future events, not under the control
of the decision maker, which will ultimately affect decision
results. States of nature should be defined so that they are
mutually exclusive and contain all possible future events that
could affect the results of all potential decisions.

Dr. C. Lightner                             2
Fayetteville State University
Decision Theory Models

Decision theory problems are generally represented as one of the
following:
– Influence Diagram
– Payoff Table
– Decision Tree

Dr. C. Lightner                       3
Fayetteville State University
Influence Diagrams

An influence diagram is a graphical device showing the
relationships among the decisions, the chance events, and the
consequences.
Squares or rectangles depict decision nodes.
Circles or ovals depict chance nodes.
Diamonds depict consequence nodes.
Lines or arcs connecting the nodes show the direction of influence.

Dr. C. Lightner                         4
Fayetteville State University
Payoff Tables

The consequence resulting from a specific combination of a
decision alternative and a state of nature is a payoff.
A table showing payoffs for all combinations of decision alternatives
and states of nature is a payoff table.
Payoffs can be expressed in terms of profit, cost, time, distance or
any other appropriate measure.

Dr. C. Lightner                         5
Fayetteville State University
Decision Trees

A decision tree is a chronological representation of the decision
problem.
Each decision tree has two types of nodes; round nodes
correspond to the states of nature while square nodes
correspond to the decision alternatives.
The branches leaving each round node represent the different
states of nature while the branches leaving each square node
represent the different decision alternatives.
At the end of each limb of a tree are the payoffs attained from the
series of branches making up that limb.

Dr. C. Lightner                            6
Fayetteville State University
Example: CAL Condominium Complex

A developer must decide how large a luxury condominium
complex to build – small, medium, or large. The
profitability of this complex depends upon the future level
of demand for the complex’s condominiums.

Dr. C. Lightner                   7
Fayetteville State University
CAL Condos: Elements of Decision Theory

States of nature: The states of nature could be defined as
low demand and high demand.
Alternatives: CAL could decide to build a small, medium,
or large condominium complex.
Payoffs: The profit for each alternative under each potential
state of nature is going to be determined.

We develop different models for this problem on the following slides.

Dr. C. Lightner                        8
Fayetteville State University
CAL Condos: Payoff Table

THIS IS A PROFIT PAYOFF TABLE

States of Nature
Alternatives            Low        High
Small                      8         8
Medium                     5       15
Large                     -11      22
(payoffs in millions of dollars)

Dr. C. Lightner           9
Fayetteville State University
CAL Condos: Decision Tree
8

8

5

Medium Complex
15

-
11

22

Dr. C. Lightner              10
Fayetteville State University
Decision Making without Probabilities

Three commonly used criteria for decision making when
probability information regarding the likelihood of the states of
nature is unavailable are:
– the optimistic approach
– the conservative approach
– the minimax regret approach.

Dr. C. Lightner                           11
Fayetteville State University
Optimistic Approach
The optimistic approach would be used by an optimistic decision
maker.
The decision with the best possible payoff is chosen.
If the payoff table was in terms of costs, the decision with the
lowest cost would be chosen.
If the payoff table was in terms of profits, the decision with the
highest profit would be chosen.

Dr. C. Lightner                          12
Fayetteville State University
Conservative Approach

The conservative approach would be used by a conservative decision maker.
For each decision the worst payoff is listed and then the decision
corresponding to the best of these worst payoffs is selected. (Hence, the
worst possible payoff is maximized.)
If the payoff was in terms of costs, the maximum costs would be determined
for each decision and then the decision corresponding to the minimum of
these maximum costs is selected. (Hence, the maximum possible cost is
minimized.)
If the payoff was in terms of profits, the minimum profits would be determined
for each decision and then the decision corresponding to the maximum of
these minimum profits is selected. (Hence, the minimum possible profit is
maximized.)

Dr. C. Lightner                                    13
Fayetteville State University
Minimax Regret Approach

1. The minimax regret approach requires the construction of a
regret table or an opportunity loss table. This is done by
calculating for each state of nature the difference between each
payoff and the best payoff for that state of nature.
2. Then, using this regret table, the maximum regret for each
possible decision is listed.
3. The decision chosen is the one corresponding to the minimum of
the maximum regrets.

Dr. C. Lightner                       14
Fayetteville State University
Solving CAL Condominiums Problem

Suppose that information regarding the probability (or likelihood) that there will be
a high or low demand is unavailable.
– A conservative or pessimistic decision maker would select the
decision alternative determined by the conservative approach.
– An optimistic decision maker would select the decision
alternative rendered by the optimistic approach.
– The minimax regret approach is generally selected by a
decision maker who reflects on decisions “after the fact”, and
complains about or “regrets” their decisions based upon the
profits that they could have made (or cheaper costs that they
could have spent) had a different decision been selected.

Dr. C. Lightner                                  15
Fayetteville State University
CAL Condos: Optimistic Decision

If the optimistic approach is selected:
STATES OF NATURE            BEST
Alternatives           Low              High       PROFIT
Small                  8                8           8
Medium                 5                15         15       Maximax
payoff
Large                 -11               22          22
Maximax
decision

Dr. C. Lightner                        16
Fayetteville State University
CAL Condos: Conservative Decision

If the conservative approach is selected:
STATES OF NATURE             WORST
Maximi        Alternatives       Low              High        PROFIT
n                                                                   Maximin
decision      Small              8                8           8         payoff

Medium             5                15          5
Large             -11               22          -11

The decision with the best profit from the column of worst profits is selected.

Dr. C. Lightner                         17
Fayetteville State University
CAL Condos: Minimax Regret Decision

If the minimax regret approach is selected:
Step 1: Determine the best payoff for each state of nature and create a regret
table.
STATES OF NATURE
Alternatives           Low         High
Small                  8           8
Medium                 5           15
Large                 -11          22

Best Profit             Best Profit
for Low                 for High
8                       22

Dr. C. Lightner                                    18
Fayetteville State University
CAL Condos: Minimax Regret Decision

If the minimax regret approach is selected:
Step 1: Create a regret table (continued).
STATES OF NATURE              For a profit payoff
table, entries in
Alternatives            Low         High              the regret table
Small                   0           14                represent profits
that could have
Medium                  3           7                 been earned.
Large                  19           0

If they knew in advanced that the demand would be low, they would have built a
small complex. Without this “psychic insight”, if they decided to build a medium
facility and the demand turned out to be low, they would regret building a medium
a small facility instead. They regret their decision by 3 million dollars.
Dr. C. Lightner                            19
Fayetteville State University
CAL Condos: Minimax Regret Decision

If the minimax regret approach is selected:
Step 2: Create a regret table (continued).
Step 3: Determine the maximum regret for each decision.
STATES OF NATURE                      Max
Alternatives          Low         High                      Regret
Small                 0           14                        14
Medium                3           7                         7
Large                19           0                         19

Regret not getting a profit
of 19 more than not making
a profit of 0.

Dr. C. Lightner                                   20
Fayetteville State University
CAL Condos: Minimax Regret Decision

If the minimax regret approach is selected:
Step 4: Select the decision with the minimum value from the column of max
regrets.
STATES OF NATURE               Max
Alternatives          Low         High               Regret
Small                 0           14                 14 Minima
Minimax      Medium                3           7                  7       x
Regret
Regret
decision
Large                19           0                  19    payoff

Dr. C. Lightner                                21
Fayetteville State University
Generic Example

Consider the following problem with three decision alternatives
and three states of nature with the following payoff table
representing costs:
States of Nature
s1      s2    s3
COST PAYOFF TABLE
d1     4.5       3        2
Decisions d2     0.5       4        1
d3      1        5        3

Dr. C. Lightner                      22
Fayetteville State University
Generic Example : Optimistic Decision

Optimistic Approach
An optimistic decision maker would use the optimistic
(maximax) approach. We choose the decision that has the best
single value in the payoff table.

Best
Decision           Cost    Maximax
Maximax               d1                 2       payoff
decision              d2                 0.5
d3                 1

Dr. C. Lightner                        23
Fayetteville State University
Generic Example: Conservative Approach

Conservative Approach
A conservative decision maker would use the conservative
(maximin) approach. List the worst payoff for each decision.
Choose the decision with the best of these worst payoffs.

Worst
Decision           Payoff   Maximin
Maximin           d1                4.5       payoff
decision
d2                4
d3                5

Dr. C. Lightner                        24
Fayetteville State University
Generic Example: Minimax Regret Decision
Minimax Regret Approach

States of Nature
For a cost payoff
s1      s2    s3                       table, entries in
the regret table
d1   4.5    3        2                      represent
overpayments
Decisions d2   0.5    4        1                      (i.e. higher costs
incurred).
d3    1     5        3

Best cost for each state of nature.

Dr. C. Lightner                                25
Fayetteville State University
Example

Minimax Regret Approach (continued)
For each decision list the maximum regret. Choose the
decision with the minimum of these values.

States of Nature             Max
s1     s2     s3             Regret

d1     4      0        1             4
Decisions d2    0       1        0             1
d3    0.5     2        2              2     Minimax
regret
Minimax
decision
Dr. C. Lightner                     26
Fayetteville State University
Decision Making with Probabilities

Expected Value Approach
– If probabilistic information regarding the states of nature is
available, one may use the expected value (EV) approach.
– Here the expected return for each decision is calculated by
summing the products of the payoff under each state of
nature and the probability of the respective state of nature
occurring.
– The decision yielding the best expected return is chosen.

Dr. C. Lightner                           27
Fayetteville State University
Expected Value of a Decision Alternative

The expected value of a decision alternative is the sum of weighted
payoffs for the decision alternative.
The expected value (EV) of decision alternative di is defined as:
N
EV( d i )   P( s j )Vij
j 1

where:     N = the number of states of nature
P(sj ) = the probability of state of nature sj
Vij = the payoff corresponding to decision
alternative di and state of nature sj

Dr. C. Lightner                        28
Fayetteville State University
Example: Burger Prince

Burger Prince Restaurant is contemplating opening a new
restaurant on Main Street. It has three different models, each
with a different seating capacity. Burger Prince estimates that
the average number of customers per hour will be 80, 100, or
120. The payoff table (profits) for the three models is on the next
slide.

Dr. C. Lightner                        29
Fayetteville State University
Example: Burger Prince

Payoff Table

Average Number of Customers Per Hour
s1 = 80 s2 = 100 s3 = 120

Model A       \$10,000 \$15,000                 \$14,000
Model B       \$ 8,000 \$18,000                \$12,000
Model C       \$ 6,000 \$16,000                \$21,000

Dr. C. Lightner                      30
Fayetteville State University
Example: Burger Prince

Expected Value Approach
Calculate the expected value for each decision. The decision
tree on the next slide can assist in this calculation. Here d1, d2, d3
represent the decision alternatives of models A, B, C, and s1, s2, s3
represent the states of nature of 80, 100, and 120.

Dr. C. Lightner                         31
Fayetteville State University
Example: Burger Prince
Payoffs
Decision Tree                                          .4
s1        10,000
s2   .2
2               s3        15,000
.4
d1                                                 14,000
s1   .4
d2                                                  8,000
1                                                 s2   .2
3                         18,000
d3                                       s3   .4
12,000
s1   .4
6,000
s2   .2
4                         16,000
s3
.4
21,000
Dr. C. Lightner                          32
Fayetteville State University
Example: Burger Prince

Expected Value For Each Decision
EMV = .4(10,000) + .2(15,000) + .4(14,000)
= \$12,600
d1           2
Model A
EMV = .4(8,000) + .2(18,000) + .4(12,000)
Model B d2                   = \$11,600
1                              3

d3            EMV = .4(6,000) + .2(16,000) + .4(21,000)
Model C
= \$14,000
4

Choose the model with largest EV, Model C.
Dr. C. Lightner                          33
Fayetteville State University
CAL Condos Revisited

Suppose market research was conducted in the community where
the complex will be built. This research allowed the company to
estimate that the probability of low demand will be 0.35, and the
probability of high demand will be 0.65. Which decision alternative
should they select.

Dr. C. Lightner                        34
Fayetteville State University
CAL Condos Revisited

STATES OF NATURE
Alternatives      Low (0.35)  High (0.65)
Small             8            8
Medium            5           15
Large            -11           22

Dr. C. Lightner           35
Fayetteville State University
CAL Condos Revisited

STATES OF NATURE
Alternatives Low       High
(0.35)   (0.65)                            Expected value (EV)
Small         8        8                                8(0.35) + 8(0.65) = 8
Medium        5       15                                5(0.35) + 15(0.65) = 11.5
Large        -11       22                              -11(0.35) + 22(0.65) = 10.45

Recall that this is a profit payoff table. Thus since the decision to build a medium
complex has the highest expected profit, this is our best decision.

Dr. C. Lightner                               36
Fayetteville State University
Expected Value of Perfect Information

Frequently information is available which can improve the
probability estimates for the states of nature.
The expected value of perfect information (EVPI) is the increase
in the expected profit that would result if one knew with certainty
which state of nature would occur.
The EVPI provides an upper bound on the expected value of any
sample or survey information.

Dr. C. Lightner                         37
Fayetteville State University
Expected Value of Perfect Information

EVPI Calculation
– Step 1:
Determine the optimal return corresponding to each state of
nature.
– Step 2:
Compute the expected value of these optimal returns.
– Step 3:
Subtract the EV of the optimal decision from the amount
determined in step (2).

Dr. C. Lightner                          38
Fayetteville State University
Example: Burger Prince

Expected Value of Perfect Information
Calculate the expected value for the optimum payoff for each
state of nature and subtract the EV of the optimal decision.

EVPI= .4(10,000) + .2(18,000) + .4(21,000) - 14,000 = \$2,000

Dr. C. Lightner                       39
Fayetteville State University
Sensitivity Analysis

Some of the quantities in a decision analysis, particularly the
probabilities, are often intelligent guesses at best.
It is important to accompany any decision analysis with a sensitivity
analysis.
Sensitivity analysis can be used to determine how changes to the
following inputs affect the recommended decision alternative:
– probabilities for the states of nature
– values of the payoffs
If a small change in the value of one of the inputs causes a change
in the recommended decision alternative, extra effort and care
should be taken in estimating the input value.

Dr. C. Lightner                        40
Fayetteville State University
Sensitivity Analysis

One approach to sensitivity analysis is to arbitrarily assign different
values to the probabilities of the states of nature and/or the payoffs
and resolve the problem. If the recommended decision changes,
then you know that the solution is sensitive to the changes.
For the special case of two states of nature, a graphical technique
can be used to determine how sensitive the solution is to the
probabilities associated with the states of nature.

Dr. C. Lightner                          41
Fayetteville State University
CAL Condos: Sensitivity Analysis

This problem has two states of nature. Previously, we stated that
CAL Condominiums estimated that the probability of future low
demand is 0.35 and 0.65 is the probability of high demand. These
probabilities yielded the recommended decision to build the medium
complex.
In order to see how sensitive this recommendation is to changing
probability values, we will let p equal the probability of low demand.
Thus (1-p) is the probability of high demand. Therefore
EV( small) = 8*p + 8*(1-p)= 8
EV( medium) = 5*p + 15*(1-p) = 15 – 10p
EV( large) = -11*p + 22*(1-p) = 22 – 33p

Dr. C. Lightner                         42
Fayetteville State University
CAL Condos: Sensitivity Analysis

Next we will plot the expected value lines for each decision by
plotting p on the x axis and EV on the y axis.
EV( small) = 8
EV( medium) = 15 – 10p
EV( large) = 22 – 33p

Dr. C. Lightner                         43
Fayetteville State University
CAL Condos: Sensitivity Analysis

25

20

EV( small)
15

10
Dr. C. Lightner           44
Fayetteville State University
CAL Condos: Sensitivity Analysis

Since CAL condominiums list payoffs are in terms of profits, we
know that the highest profits is desirable.
Look over the entire range of p (p=0 to p=1) and determine the
range over which each decision yields the highest profits.

Dr. C. Lightner                          45
Fayetteville State University
CAL Condos: Sensitivity Analysis

25

20

EV( small)
15

10
B1                             B2
Dr. C. Lightner                             46
Fayetteville State University
CAL Condos: Sensitivity Analysis

Do not estimate the values of B1 or B2 (the points where the intersection of lines
occur). Determine the exact intersection points.
B1 is the point where the EV( large) line intersects with the EV( medium) line:
To find this point set these two lines equal to each other and solve for p.
22-33p= 15-10p
7= 23p
p=7/23= 0.3403 So B1 equals 0.3403

B2 is the point where the EV( medium) line intersects with the EV( small) line:
15-10p = 8
7 = 10p
p = 0.7         So B2 equals 0.7

Dr. C. Lightner                                    47
Fayetteville State University
CAL Condos: Sensitivity Analysis

25

20

EV( small)
15

10
0.3403                         0.7
Dr. C. Lightner                              48
Fayetteville State University
CAL Condos: Sensitivity Analysis

From the graph we see that if the probability of low demand (p) is
between 0 and 0.3403, we recommend building a large complex.
From the graph we see that if the probability of low demand (p) is
between 0.3403 and 0.7, we recommend building a medium
complex.
From the graph we see that if the probability of low demand (p) is
between 0.7 and 1, we recommend building a large complex.

From this sensitivity analysis we see that if CAL Condos estimate of
0.35 for the probability of low demand was slightly lower, the
recommended decision would change.

Dr. C. Lightner                        49
Fayetteville State University
End of Chapter 14

examples and detailed explanations
of all topics discussed in these notes.

Dr. C. Lightner           50
Fayetteville State University

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