### Decision Analysis - MCST-CS

```Decision Analysis
Chapter 12
12-1
Chapter Topics
■ Components of Decision Making
■ Decision Making without Probabilities
■ Decision Making with Probabilities
■ Decision Analysis with Additional Information
■ Utility
12-2
Decision Analysis
Overview
 Previous chapters used an assumption of certainty with regards to
problem parameters.
 This chapter relaxes the certainty assumption
 Two categories of decision situations:
 Probabilities can be assigned to future occurrences
 Probabilities cannot be assigned to future occurrences
12-3
Decision Analysis
Components of Decision Making
■ A state of nature is an actual event that may occur in the future.
■ A payoff table is a means of organizing a decision situation,
presenting the payoffs from different decisions given the various
states of nature.
Table 12.1 Payoff table
12-4
Decision Analysis
Decision Making Without Probabilities
Figure 12.1 Decision
situation with real estate
investment alternatives
12-5
Decision Analysis
Decision Making without Probabilities
Table 12.2 Payoff table for the real estate investments
Decision-Making Criteria
maximax
maximin
minimax
minimax regret
Hurwicz
equal likelihood
12-6
Decision Making without Probabilities
Maximax Criterion
In the maximax criterion the decision maker selects the decision
that will result in the maximum of maximum payoffs; an
optimistic criterion.
Table 12.3
Payoff table illustrating a maximax decision
12-7
Decision Making without Probabilities
Maximin Criterion
In the maximin criterion the decision maker selects the decision
that will reflect the maximum of the minimum payoffs; a
pessimistic criterion.
Table 12.4 Payoff table illustrating a maximin decision
12-8
Decision Making without Probabilities
Minimax Regret Criterion
 Regret is the difference between the payoff from the best
decision and all other decision payoffs.
 Example: under the Good Economic Conditions state of nature,
the best payoff is \$100,000. The manager’s regret for choosing
the Warehouse alternative is \$100,000-\$30,000=\$70,000
Table 12.5 Regret table
12-9
Decision Making without Probabilities
Minimax Regret Criterion
 The manager calculates regrets for all alternatives under each
state of nature. Then the manager identifies the maximum
regret for each alternative.
 Finally, the manager attempts to avoid regret by selecting the
decision alternative that minimizes the maximum regret.
Table 12.6
Regret table illustrating the minimax regret decision
12-10
Decision Making without Probabilities
Hurwicz Criterion
 The Hurwicz criterion is a compromise between the maximax
and maximin criteria.
 A coefficient of optimism, , is a measure of the decision
maker’s optimism.
 The Hurwicz criterion multiplies the best payoff by  and the
worst payoff by 1- , for each decision, and the best result is
selected. Here,  = 0.4.
Decision
Apartment building
Values
\$50,000(.4) + 30,000(.6) = 38,000
Office building
\$100,000(.4) - 40,000(.6) = 16,000
Warehouse
\$30,000(.4) + 10,000(.6) = 18,000
12-11
Decision Making without Probabilities
Equal Likelihood Criterion
The equal likelihood ( or Laplace) criterion multiplies the
decision payoff for each state of nature by an equal weight, thus
assuming that the states of nature are equally likely to occur.
Decision
Apartment building
Values
\$50,000(.5) + 30,000(.5) = 40,000
Office building
\$100,000(.5) - 40,000(.5) = 30,000
Warehouse
\$30,000(.5) + 10,000(.5) = 20,000
12-12
Decision Making without Probabilities
Summary of Criteria Results
■ A dominant decision is one that has a better payoff than another
decision under each state of nature.
■ The appropriate criterion is dependent on the “risk” personality
and philosophy of the decision maker.
Criterion
Decision (Purchase)
Maximax
Office building
Maximin
Apartment building
Minimax regret
Apartment building
Hurwicz
Apartment building
Equal likelihood
Apartment building
12-13
Decision Making without Probabilities
Solution with QM for Windows (1 of 3)
Exhibit 12.1
12-14
Decision Making without Probabilities
Solution with QM for Windows (2 of 3)
Equal likelihood weight
Exhibit 12.2
12-15
Decision Making without Probabilities
Solution with QM for Windows (3 of 3)
Exhibit 12.3
12-16
Decision Making without Probabilities
Solution with Excel
=MIN(C7,D7)
=MAX(E7,E9)
=MAX(F7:F9)
=MAX(C18,D18)
=MAX(C7:C9)-C9
=C7*C25+D7*C26
=C7*0.5+D7*0.5
Exhibit 12.4
12-17
Decision Making with Probabilities
Expected Value
 Expected value is computed by multiplying each decision
outcome under each state of nature by the probability of its
occurrence.
Table 12.7 Payoff table with probabilities for states of nature
EV(Apartment) = \$50,000(.6) + 30,000(.4) = \$42,000
EV(Office) = \$100,000(.6) - 40,000(.4) = \$44,000
EV(Warehouse) = \$30,000(.6) + 10,000(.4) = \$22,000
12-18
Decision Making with Probabilities
Expected Opportunity Loss
■ The expected opportunity loss is the expected value of the
regret for each decision.
■ The expected value and expected opportunity loss criterion
result in the same decision.
EOL(Apartment) = \$50,000(.6) + 0(.4) = 30,000
EOL(Office) = \$0(.6) + 70,000(.4) = 28,000
EOL(Warehouse) = \$70,000(.6) + 20,000(.4) = 50,000
Table 12.8 Regret table with probabilities for states of nature
12-19
Expected Value Problems
Solution with QM for Windows
Expected values
Exhibit 12.5
12-20
Expected Value Problems
Solution with Excel and Excel QM (1 of 2)
Expected value for
apartment building
Exhibit 12.6
12-21
Expected Value Problems
Solution with Excel and Excel QM (2 of 2)
Exhibit 12.7
12-22
Decision Making with Probabilities
Expected Value of Perfect Information
■ The expected value of perfect information (EVPI) is the
maximum amount a decision maker would pay for additional
information.
■ EVPI equals the expected value given perfect information
minus the expected value without perfect information.
■ EVPI equals the expected opportunity loss (EOL) for the best
decision.
12-23
Decision Making with Probabilities
EVPI Example (1 of 2)
Table 12.9 Payoff table with decisions, given perfect information
12-24
Decision Making with Probabilities
EVPI Example (2 of 2)
■ Decision with perfect information:
\$100,000(.60) + 30,000(.40) = \$72,000
■ Decision without perfect information:
EV(office) = \$100,000(.60) - 40,000(.40) = \$44,000
EVPI = \$72,000 - 44,000 = \$28,000
EOL(office) = \$0(.60) + 70,000(.4) = \$28,000
12-25
Decision Making with Probabilities
EVPI with QM for Windows
The expected value, given
perfect information, in Cell F12
=MAX(E7:E9)
=F12-F11
Exhibit 12.8
12-26
Decision Making with Probabilities
Decision Trees (1 of 4)
A decision tree is a diagram consisting of decision nodes
(represented as squares), probability nodes (circles), and
decision alternatives (branches).
Table 12.10 Payoff table for real estate investment example
12-27
Decision Making with Probabilities
Decision Trees (2 of 4)
Figure 12.2 Decision tree for real estate investment example
12-28
Decision Making with Probabilities
Decision Trees (3 of 4)
■ The expected value is computed at each probability node:
EV(node 2) = .60(\$50,000) + .40(30,000) = \$42,000
EV(node 3) = .60(\$100,000) + .40(-40,000) = \$44,000
EV(node 4) = .60(\$30,000) + .40(10,000) = \$22,000
■ Branches with the greatest expected value are selected.
12-29
Decision Making with Probabilities
Decision Trees (4 of 4)
Figure 12.3 Decision tree with expected value at probability nodes
12-30
Decision Making with Probabilities
Decision Trees with QM for Windows
Number of branches
from node 1
1 to 2, 3, and 4
Exhibit 12.9
12-31
Decision Making with Probabilities
Decision Trees with Excel and TreePlan (1 of 4)
Exhibit 12.10
12-32
Decision Making with Probabilities
Decision Trees with Excel and TreePlan (2 of 4)
To create another branch, click
“B5,” then the “Decision Tree”
Invoke TreePlan from
Exhibit 12.11
12-33
Decision Making with Probabilities
Decision Trees with Excel and TreePlan (3 of 4)
Click on cell “F3,”
then “Decision Tree”
Select “Change to
two new branches
Exhibit 12.12
12-34
Decision Making with Probabilities
Decision Trees with Excel and TreePlan (4 of 4)
values in these cells in column H
Exhibit 12.13
These cells contain decision tree
formulas; do not type in these
cells in columns E and I
12-35
Sequential Decision Tree Analysis
Solution with QM for Windows
Cell A16 contains
the expected value
of \$44,000
Exhibit 12.14
12-36
Decision Making with Probabilities
Sequential Decision Trees (1 of 4)
■ A sequential decision tree is used to illustrate a situation
requiring a series of decisions.
■ Used where a payoff table, limited to a single decision, cannot
be used.
■ The next slide shows the real estate investment example
modified to encompass a ten-year period in which several
12-37
Decision Making with Probabilities
Sequential Decision Trees (2 of 4)
Figure 12.4 Sequential decision tree
12-38
Decision Making with Probabilities
Sequential Decision Trees (3 of 4)
■ Expected value of apartment building is:
\$1,290,000-800,000 = \$490,000
■ Expected value if land is purchased is:
\$1,360,000-200,000 = \$1,160,000
■ The decision is to purchase land; it has the highest net expected
value of \$1,160,000.
12-39
Decision Making with Probabilities
Sequential Decision Trees (4 of 4)
Figure 12.5 Sequential decision tree with nodal expected values
12-40
Sequential Decision Tree Analysis
Solution with Excel QM
Exhibit 12.15
12-41
Sequential Decision Tree Analysis
Solution with TreePlan
Exhibit 12.16
12-42
Bayesian Analysis (1 of 3)
■ Bayesian analysis uses additional information to alter the
marginal probability of the occurrence of an event.
■ In the real estate investment example, using the expected value
criterion, the best decision was to purchase the office building
with an expected value of \$444,000, and EVPI of \$28,000.
Table 12.11 Payoff table for real estate investment
12-43
Bayesian Analysis (2 of 3)
■ A conditional probability is the probability that an event will
occur given that another event has already occurred.
■ An economic analyst provides additional information for the
real estate investment decision, forming conditional
probabilities:
g = good economic conditions
p = poor economic conditions
P = positive economic report
N = negative economic report
P(Pg) = .80
P(NG) = .20
P(Pp) = .10
P(Np) = .90
12-44
Bayesian Analysis (3 of 3)
■ A posterior probability is the altered marginal probability of an
■ Prior probabilities for good or poor economic conditions in the
real estate decision:
P(g) = .60; P(p) = .40
■ Posterior probabilities by Bayes’ rule:
(gP) = P(PG)P(g)/[P(Pg)P(g) + P(Pp)P(p)]
= (.80)(.60)/[(.80)(.60) + (.10)(.40)] = .923
■ Posterior (revised) probabilities for decision:
P(gN) = .250
P(pP) = .077
P(pN) = .750
12-45
Decision Trees with Posterior Probabilities (1 of 4)
Decision trees with posterior probabilities differ from earlier
versions in that:
■ Two new branches at the beginning of the tree represent
report outcomes.
■ Probabilities of each state of nature are posterior
probabilities from Bayes’ rule.
12-46
Decision Trees with Posterior Probabilities (2 of 4)
Figure 12.6 Decision tree with posterior probabilities
12-47
Decision Trees with Posterior Probabilities (3 of 4)
EV (apartment building) = \$50,000(.923) + 30,000(.077)
= \$48,460
EV (strategy) = \$89,220(.52) + 35,000(.48) = \$63,194
12-48
Decision Trees with Posterior Probabilities (4 of 4)
Figure 12.7 Decision tree analysis for real estate investment
12-49
Computing Posterior Probabilities with Tables
Table 12.12 Computation of posterior probabilities
12-50
Computing Posterior Probabilities with Excel
Exhibit 12.17
12-51
Expected Value of Sample Information
■ The expected value of sample information (EVSI) is the
difference between the expected value with and without
information:
For example problem, EVSI = \$63,194 - 44,000 = \$19,194
■ The efficiency of sample information is the ratio of the
expected value of sample information to the expected value of
perfect information:
efficiency = EVSI /EVPI = \$19,194/ 28,000 = .68
12-52
Utility (1 of 2)
Table 12.13 Payoff table for auto insurance example
12-53
Utility (2 of 2)
Expected Cost (insurance) = .992(\$500) + .008(500) = \$500
Expected Cost (no insurance) = .992(\$0) + .008(10,000) = \$80
The decision should be do not purchase insurance, but people
almost always do purchase insurance.
■ Utility is a measure of personal satisfaction derived from money.
■ Utiles are units of subjective measures of utility.
■ Risk averters forgo a high expected value to avoid a lowprobability disaster.
■ Risk takers take a chance for a bonanza on a very lowprobability event in lieu of a sure thing.
12-54
Decision Analysis
Example Problem Solution (1 of 9)
A corporate raider contemplates the future of a recent acquisition.
Three alternatives are being considered in two states of nature. The
payoff table is below.
States of Nature
Decision
Expand
Maintain Status Quo
Sell now
Good Foreign Competitive
Conditions
\$ 800,000
1,300,000
320,000
Poor Foreign Competitive
Conditions
\$ 500,000
-150,000
320,000
12-55
Decision Analysis
Example Problem Solution (2 of 9)
a. Determine the best decision without probabilities using the 5
criteria of the chapter.
b. Determine best decision with probabilities assuming .70
probability of good conditions, .30 of poor conditions. Use
expected value and expected opportunity loss criteria.
c. Compute expected value of perfect information.
d. Develop a decision tree with expected value at the nodes.
e. Given the following, P(Pg) = .70, P(Ng) = .30, P(Pp) = 20,
P(Np) = .80, determine posterior probabilities using Bayes’
rule.
f. Perform a decision tree analysis using the posterior probability
obtained in part e.
12-56
Decision Analysis
Example Problem Solution (3 of 9)
Step 1 (part a): Determine decisions without probabilities.
Maximax Decision: Maintain status quo
Decisions
Maximum Payoffs
Expand
Status quo
Sell
\$800,000
1,300,000 (maximum)
320,000
Maximin Decision: Expand
Decisions
Minimum Payoffs
Expand
Status quo
Sell
\$500,000 (maximum)
-150,000
320,000
12-57
Decision Analysis
Example Problem Solution (4 of 9)
Minimax Regret Decision: Expand
Decisions
Maximum Regrets
Expand
\$500,000 (minimum)
Status quo
650,000
Sell
980,000
Hurwicz ( = .3) Decision: Expand
Expand
\$800,000(.3) + 500,000(.7) = \$590,000
Status quo
\$1,300,000(.3) - 150,000(.7) = \$285,000
Sell
\$320,000(.3) + 320,000(.7) = \$320,000
12-58
Decision Analysis
Example Problem Solution (5 of 9)
Equal Likelihood Decision: Expand
Expand
Status quo
Sell
\$800,000(.5) + 500,000(.5) = \$650,000
\$1,300,000(.5) - 150,000(.5) = \$575,000
\$320,000(.5) + 320,000(.5) = \$320,000
Step 2 (part b): Determine Decisions with EV and EOL.
Expected value decision: Maintain status quo
Expand
Status quo
Sell
\$800,000(.7) + 500,000(.3) = \$710,000
\$1,300,000(.7) - 150,000(.3) = \$865,000
\$320,000(.7) + 320,000(.3) = \$320,000
12-59
Decision Analysis
Example Problem Solution (6 of 9)
Expected opportunity loss decision: Maintain status quo
Expand
\$500,000(.7) + 0(.3) = \$350,000
Status quo
0(.7) + 650,000(.3) = \$195,000
Sell
\$980,000(.7) + 180,000(.3) = \$740,000
Step 3 (part c): Compute EVPI.
EV given perfect information =
1,300,000(.7) + 500,000(.3) = \$1,060,000
EV without perfect information =
\$1,300,000(.7) - 150,000(.3) = \$865,000
EVPI = \$1,060,000 - 865,000 = \$195,000
12-60
Decision Analysis
Example Problem Solution (7 of 9)
Step 4 (part d): Develop a decision tree.
12-61
Decision Analysis
Example Problem Solution (8 of 9)
Step 5 (part e): Determine posterior probabilities.
P(gP) = P(Pg)P(g)/[P(Pg)P(g) + P(Pp)P(p)]
= (.70)(.70)/[(.70)(.70) + (.20)(.30)] = .891
P(pP) = .109
P(gN) = P(Ng)P(g)/[P(Ng)P(g) + P(Np)P(p)]
= (.30)(.70)/[(.30)(.70) + (.80)(.30)] = .467
P(pN) = .533