CH_12_DECISION_ANALYSIS

Decision Analysis

Chapter 12 12-1

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Chapter Topics

■ Components of Decision Making
■ Decision Making without Probabilities
■ Decision Making with Probabilities
■ Decision Analysis with Additional Information
■ Utility

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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.

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12-3

Decision Analysis
Decision Making Without Probabilities

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12-4

Decision Analysis
Decision Making without Probabilities

Decision-Making Criteria Table 12.2

maximax maximin minimax
equal likelihood
minimax regret Hurwicz
12-5
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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 12-6

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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 12-7
Decision

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Decision Making without Probabilities
Minimax Regret Criterion

Regret is the difference between the payoff from the best

decision and all other decision payoffs.

The decision maker attempts to avoid regret by selecting the

decision alternative that minimizes the maximum regret.

Table 12.6 Regret Table Illustrating the Minimax Regret Decision 12-8

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Decision Making without Probabilities
Hurwicz Criterion

The Hurwicz criterion is a compromise between the maximax

and maximin criterion.

A coefficient of optimism, α, is a measure of the decision

maker’s optimism.

The Hurwicz criterion multiplies the best payoff by α and the
worst payoff by 1- α., for each decision, and the best result is

selected.

Decision Values

Apartment building $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

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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 Values

Apartment building $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

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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

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Decision Making without Probabilities
Solution with QM for Windows (1 of 3)

Exhibit 12.1

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Decision Making without Probabilities
Solution with QM for Windows (2 of 3)

Exhibit 12.2

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Decision Making without Probabilities
Solution with QM for Windows (3 of 3)

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12-14

Decision Making without Probabilities
Solution with Excel

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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.

EV(Apartment) = $50,000(.6) + 30,000(.4) = 42,000 Table 12.7
EV(Office) = $100,000(.6) - 40,000(.4) = 44,000
EV(Warehouse) = $30,000(.6) + 10,000(.4) = 22,000 12-16

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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 Table 12.8
EOL(Office) = $0(.6) + 70,000(.4) = 28,000
EOL(Warehouse) = $70,000(.6) + 20,000(.4) = 50,000 12-17

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Expected Value Problems
Solution with QM for Windows

Exhibit 12.5

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Expected Value Problems
Solution with Excel and Excel QM (1 of 2)

Exhibit 12.6

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Expected Value Problems
Solution with Excel and Excel QM (2 of 2)

Exhibit 12.7

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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.

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Decision Making with Probabilities
EVPI Example (1 of 2)

Table 12.9 Payoff Table with Decisions, Given Perfect Information

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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

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Decision Making with Probabilities
EVPI with QM for Windows

Exhibit 12.8

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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

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Decision Making with Probabilities
Decision Trees (2 of 4)

Figure 12.2 Decision Tree for Real Estate Investment Example 12-26

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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.

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Decision Making with Probabilities
Decision Trees (4 of 4)

Figure 12.3 Decision Tree with Expected Value at Probability Nodes

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Decision Making with Probabilities
Decision Trees with QM for Windows

Exhibit 12.9

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Decision Making with Probabilities
Decision Trees with Excel and TreePlan (1 of 4)

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12-30

Decision Making with Probabilities
Decision Trees with Excel and TreePlan (2 of 4)

Exhibit 12.11

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Decision Making with Probabilities
Decision Trees with Excel and TreePlan (3 of 4)

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12-32

Decision Making with Probabilities
Decision Trees with Excel and TreePlan (4 of 4)

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12-33

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.

■ Real estate investment example modified to encompass a ten-
year period in which several decisions must be made:

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Decision Making with Probabilities
Sequential Decision Trees (2 of 4)

Figure 12.4 Sequential Decision Tree 12-35

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Decision Making with Probabilities
Sequential Decision Trees (3 of 4)

■ Decision is to purchase land; highest net expected value
($1,160,000).

■ Payoff of the decision is $1,160,000.

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Decision Making with Probabilities
Sequential Decision Trees (4 of 4)

Figure 12.5 Sequential Decision Tree with Nodal Expected Values

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Sequential Decision Tree Analysis
Solution with QM for Windows

Exhibit 12.14

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Sequential Decision Tree Analysis
Solution with Excel and TreePlan

Exhibit 12.15

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Decision Analysis with Additional Information

Bayesian Analysis (1 of 3)

■ Bayesian analysis uses additional information to alter the
marginal probability of the occurrence of an event.

■ In real estate investment example, using expected value
criterion, best decision was to purchase office building with
expected value of $444,000, and EVPI of $28,000.

Table 12.11

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Decision Analysis with Additional Information
Bayesian Analysis (2 of 3)

■ A conditional probability is the probability that an event will

occur given that another event has already occurred.

■ Economic analyst provides additional information for 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

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Decision Analysis with Additional Information
Bayesian Analysis (3 of 3)

■ A posterior probability is the altered marginal probability of an

event based on additional information.

■ Prior probabilities for good or poor economic conditions in 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) = .077P(pN) = .750

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Decision Analysis with Additional Information
Decision Trees with Posterior Probabilities (1 of 4)

Decision tree with posterior probabilities differ from earlier
versions in that:
■ Two new branches at beginning of tree represent report

outcomes.

■ Probabilities of each state of nature are posterior
probabilities from Bayes’ rule.

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Decision Analysis with Additional Information
Decision Trees with Posterior Probabilities (2 of 4)

Figure 12.6 Decision Tree with Posterior Probabilities 12-44

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Decision Analysis with Additional Information
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

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Decision Analysis with Additional Information
Decision Trees with Posterior Probabilities (4 of 4)

Figure 12.7 Decision Tree Analysis 12-46

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Decision Analysis with Additional Information
Computing Posterior Probabilities with Tables

Table 12.12 Computation of Posterior Probabilities 12-47

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Decision Analysis with Additional Information
Computing Posterior Probabilities with Excel

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12-48

Decision Analysis with Additional Information
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

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Decision Analysis with Additional Information
Utility (1 of 2)

Table 12.13 Payoff Table for Auto Insurance 12-50
Example

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