Decision Analysis

1
Chapter 12 Decision Analysis
2
Chapter Topics

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

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
4
Decision AnalysisDecision Making Without
Probabilities
Figure 12.1
5
Decision Analysis Decision Making without
Probabilities

• Decision situation
• Decision-Making Criteria maximax, maximin,
  minimax, minimax regret, Hurwicz, and equal
  likelihood

Table 12.2 Payoff Table for the Real Estate
Investments

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
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
8
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
9
Decision Making without Probabilities Hurwicz
Criterion

• The Hurwicz criterion is a compromise between the
  maximax and maximin criterion.
• A coefficient of optimism, ?, is a measure of the
  decision makers optimism.
• The Hurwicz criterion multiplies the best payoff
  by ? and the worst payoff by 1- ?., 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

10
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

11
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
Decision Making without Probabilities Solution
with QM for Windows (1 of 3)
Exhibit 12.1
13
Decision Making without Probabilities Solution
with QM for Windows (2 of 3)
Exhibit 12.2
14
Decision Making without Probabilities Solution
with QM for Windows (3 of 3)
Exhibit 12.3
15
Decision Making without Probabilities Solution
with Excel
Exhibit 12.4
16
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
• EV(Office) 100,000(.6) - 40,000(.4) 44,000
• EV(Warehouse) 30,000(.6) 10,000(.4)
  22,000

Table 12.7 Payoff table with Probabilities for
States of Nature
17
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 (Opportunity Loss) Table
with Probabilities for States of Nature
18
Expected Value Problems Solution with QM for
Windows
Exhibit 12.5
19
Expected Value Problems Solution with Excel and
Excel QM (1 of 2)
Exhibit 12.6
20
Expected Value Problems Solution with Excel and
Excel QM (2 of 2)
Exhibit 12.7
21
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.

22
Decision Making with Probabilities EVPI Example
(1 of 2)
Table 12.9 Payoff Table with Decisions, Given
Perfect Information
23
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

24
Decision Making with Probabilities EVPI with QM
for Windows
Exhibit 12.8
25
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
26
Decision Making with Probabilities Decision Trees
(2 of 4)
Figure 12.1 Decision Tree for Real Estate
Investment Example
27
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.

28
Decision Making with Probabilities Decision Trees
(4 of 4)
Figure 12.2 Decision Tree with Expected Value at
Probability Nodes
29
Decision Making with Probabilities Decision Trees
with QM for Windows
Exhibit 12.9
30
Decision Making with Probabilities Decision Trees
with Excel and TreePlan (1 of 4)
Exhibit 12.10
31
Decision Making with Probabilities Decision Trees
with Excel and TreePlan (2 of 4)
Exhibit 12.11
32
Decision Making with Probabilities Decision Trees
with Excel and TreePlan (3 of 4)
Exhibit 12.12
33
Decision Making with Probabilities Decision Trees
with Excel and TreePlan (4 of 4)
Exhibit 12.13
34
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

35
Decision Making with Probabilities Sequential
Decision Trees (2 of 4)
Figure 12.3 Sequential Decision Tree
36
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.

37
Decision Making with Probabilities Sequential
Decision Trees (4 of 4)
Figure 12.4 Sequential Decision Tree with Nodal
Expected Values
38
Sequential Decision Tree Analysis Solution with
QM for Windows
Exhibit 12.14
39
Sequential Decision Tree Analysis Solution with
Excel and TreePlan
Exhibit 12.15
40
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 44,000,
  and EVPI of 28,000.

Table 12.11 Payoff Table for the Real Estate
Investment Example
41
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

42
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
• P(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

43
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.

44
Decision Analysis with Additional
Information Decision Trees with Posterior
Probabilities (2 of 4)
Figure 12.5 Decision Tree with Posterior
Probabilities
45
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

46
Decision Analysis with Additional
Information Decision Trees with Posterior
Probabilities (4 of 4)
Figure 12.6 Decision Tree Analysis
47
Decision Analysis with Additional
Information Computing Posterior Probabilities
with Tables
Table 12.12 Computation of Posterior
Probabilities
48
Decision Analysis with Additional Information
Computing Posterior Probabilities with Excel
Exhibit 12.16
49
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

50
Decision Analysis with Additional
Information Utility (1 of 2)
Table 12.13 Payoff Table for Auto Insurance
Example
51
Decision Analysis with Additional
Information Utility (2 of 2)

• Expected Cost (insurance) .992(500)
  .008(500) 500
• Expected Cost (no insurance) .992(0)
  .008(10,000) 80
• 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 low-probability disaster.
• Risk takers take a chance for a bonanza on a very
  low-probability event in lieu of a sure thing.

52
Decision Analysis Example Problem Solution (1 of
9)
53
Decision Analysis Example Problem Solution (2 of
9)

• Determine the best decision without
  probabilities using the 5 criteria of the
  chapter.
• Determine best decision with probabilities
  assuming .70 probability of good conditions, .30
  of poor conditions. Use expected value and
  expected opportunity loss criteria.
• Compute expected value of perfect information.
• Develop a decision tree with expected value at
  the nodes.
• Given following, P(P?g) .70, P(N?g) .30,
  P(P?p) .20, P(N?p) .80, determine posterior
  probabilities using Bayes rule.
• Perform a decision tree analysis using the
  posterior probability obtained in part e.

54
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
800,000 Status quo 1,300,000 (maximum) Sell
320,000 Maximin Decision Expand Decisions Mini
mum Payoffs Expand 500,000 (maximum) Status
quo -150,000 Sell 320,000
55
Decision Analysis Example Problem Solution (4 of
9)
Minimax Regret Decision Expand Decisions Maximu
m 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
56
Decision Analysis Example Problem Solution (5 of
9)
Equal Likelihood Decision Expand Expand
800,000(.5) 500,000(.5) 650,000 Status
quo 1,300,000(.5) - 150,000(.5)
575,000 Sell 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 800,000(.7)
500,000(.3) 710,000 Status quo
1,300,000(.7) - 150,000(.3) 865,000 Sell
320,000(.7) 320,000(.3) 320,000
57
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
58
Decision Analysis Example Problem Solution (7 of
9)
Step 4 (part d) Develop a decision tree.

59
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
60
Decision Analysis Example Problem Solution (9 of
9)
Step 6 (part f) Decision tree analysis.
61
Assignment for Chapter 12 Problems 5,
10,17,19,28, and 35