Decision Making

1. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc. Chap 17-1
Chapter 17
Decision Making
Basic Business Statistics
11th Edition

2. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-2
Learning Objectives
In this chapter, you learn:
 To use payoff tables and decision trees to
evaluate alternative courses of action
 To use several criteria to select an alternative
course of action
 To use Bayes’ theorem to revise probabilities in
light of sample information
 About the concept of utility

3. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-3
Steps in Decision Making
 List Alternative Courses of Action
 Choices or actions
 List Uncertain Events
 Possible events or outcomes
 Determine ‘Payoffs’
 Associate a Payoff with Each Choice/Event
combination
 Adopt Decision Criteria
 Evaluate Criteria for Selecting the Best Course of
Action

4. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-4
List Possible Actions or Events
Payoff Table Decision Tree
Two Methods
of Listing

5. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-5
A Payoff Table
A payoff table shows alternatives,
states of nature, and payoffs
States of Nature
(Events)
Profit in $1,000’s
Investment Choice (Action)
Large
Factory
Average
Factory
Small
Factory
Strong Economy
Stable Economy
Weak Economy
200
50
-120
90
120
-30
40
30
20

6. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-6
Sample Decision Tree
Large factory
Small factory
Average factory
Strong Economy
Stable Economy
Weak Economy
Strong Economy
Stable Economy
Weak Economy
Strong Economy
Stable Economy
Weak Economy
Payoffs
200
50
-120
40
30
20
90
120
-30

7. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-7
Opportunity Loss
State Of Nature
(Events)
Profit in $1,000’s
(Action)
Large
Factory
Average
Factory
Small
Factory
Strong Economy
Stable Economy
Weak Economy
200
50
-120
90
120
-30
40
30
20
The action “Average factory” has payoff 90 for “Strong Economy”. Given
“Strong Economy”, the choice of “Large factory” would have given a
payoff of 200, or 110 higher. Opportunity loss = 110 for this cell.
Opportunity loss is the difference between an actual
payoff for an action and the highest possible payoff,
given a particular event
Payoff
Table

8. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-8
Opportunity Loss
States of Nature
(Events)
Profit in $1,000’s
Investment Choice (Action)
Large
Factory
Average
Factory
Small
Factory
Strong Economy
Stable Economy
Weak Economy
200
50
-120
90
120
-30
40
30
20
(continued)
States of Nature
(Events)
Opportunity Loss in $1,000’s
Investment Choice (Action)
Large
Factory
Average
Factory
Small
Factory
Strong Economy
Stable Economy
Weak Economy
0
70
140
110
0
50
160
90
0
Payoff
Table
Opportunity
Loss Table

9. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-9
Decision Criteria
 Maximax
 An optimistic decision criteria
 Maximin
 A pessimistic decision criteria
 Expected Monetary Value (EMV(j))
 The expected profit for taking action j
 Expected Opportunity Loss (EOL(j))
 The expected opportunity loss for taking action j
 Expected Value of Perfect Information (EVPI)
 The expected opportunity loss from the best decision
 Return to Risk Ratio
 Takes into account the variability in payoffs

10. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-10
Maximax Solution
States of Nature
(Events)
Profit in $1,000’s
Investment Choice (Action)
Large
Factory
Average
Factory
Small
Factory
Strong Economy
Stable Economy
Weak Economy
200
50
-120
90
120
-30
40
30
20
Maximum payoff for Large Factory is 200
Maximum payoff for Average Factory is 120
Maximum payoff for Small Factory is 40
Maximax Decision: Build the Large Factory because 200 is the maximum

11. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-11
Maximin Solution
States of Nature
(Events)
Profit in $1,000’s
Investment Choice (Action)
Large
Factory
Average
Factory
Small
Factory
Strong Economy
Stable Economy
Weak Economy
200
50
-120
90
120
-30
40
30
20
Minimum payoff for Large Factory is -120
Minimum payoff for Average Factory is -30
Minimum payoff for Small Factory is 20
Maximin Decision: Build the Small Factory because 20 is the maximum

12. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-12
Expected Monetary Value
Solution
 The expected monetary value is the weighted
average payoff, given specified probabilities for
each event


N
1i
iijPx)j(EMV
Where EMV(j) = expected monetary value of action j
xij = payoff for action j when event i occurs
Pi = probability of event i
Goal: Maximize expected value

13. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-13
 The expected value is the weighted average
payoff, given specified probabilities for each event
States of Nature
(Events)
Profit in $1,000’s
Investment Choice (Action)
Large
Factory
Average
Factory
Small
Factory
Strong Economy (.3)
Stable Economy (.5)
Weak Economy (.2)
200
50
-120
90
120
-30
40
30
20
Suppose these probabilities have been
assessed for these three events
(continued)
Expected Monetary Value
Solution

14. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-14
States of Nature
(Events)
Profit in $1,000’s
Investment Choice (Action)
Large
Factory
Average
Factory
Small
Factory
Strong Economy (.3)
Stable Economy (.5)
Weak Economy (.2)
200
50
-120
90
120
-30
40
30
20
Example:
EMV (Average factory) =
(90)(.3) +
(120)(.5) +
(-30)(.2) = 81
Expected Value (EMV) 61 81 31
Maximize expected value by
choosing Average factory
(continued)
Payoff Table:
Goal: Maximize expected value
Expected Monetary Value
Solution

15. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-15
Decision Trees Are Another Way To
Display & Analyze The Same Information
 A Decision tree shows a decision problem,
beginning with the initial decision and ending
will all possible outcomes and payoffs.
Use a square to denote decision nodes
Use a circle to denote uncertain events

16. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-16
Add Probabilities and Payoffs
Large factory
Small factory
Decision
Average factory
Uncertain Events
Strong Economy
Stable Economy
Weak Economy
Strong Economy
Stable Economy
Weak Economy
Strong Economy
Stable Economy
Weak Economy
(continued)
PayoffsProbabilities
200
50
-120
40
30
20
90
120
-30
(.3)
(.5)
(.2)
(.3)
(.5)
(.2)
(.3)
(.5)
(.2)

17. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-17
Fold Back the Tree
Large factory
Small factory
Average factory
Strong Economy
Stable Economy
Weak Economy
Strong Economy
Stable Economy
Weak Economy
Strong Economy
Stable Economy
Weak Economy
200
50
-120
40
30
20
90
120
-30
(.3)
(.5)
(.2)
(.3)
(.5)
(.2)
(.3)
(.5)
(.2)
EMV=200(.3)+50(.5)+(-120)(.2)=61
EMV=90(.3)+120(.5)+(-30)(.2)=81
EMV=40(.3)+30(.5)+20(.2)=31

18. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-18
Make the Decision
Large factory
Small factory
Average factory
Strong Economy
Stable Economy
Weak Economy
Strong Economy
Stable Economy
Weak Economy
Strong Economy
Stable Economy
Weak Economy
200
50
-120
40
30
20
90
120
-30
(.3)
(.5)
(.2)
(.3)
(.5)
(.2)
(.3)
(.5)
(.2)
EV=61
EV=81
EV=31
Maximum
EMV=81

19. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-19
Expected Opportunity Loss
Solution
 The expected opportunity loss is the weighted
average loss, given specified probabilities for
each event


N
1i
iijPL)j(EOL
Where: EOL(j) = expected opportunity loss of action j
Lij = opportunity loss for action j when event i occurs
Pi = probability of event i
Goal: Minimize expected opportunity loss

20. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-20
Expected Opportunity Loss
Solution
States of Nature
(Events)
Opportunity Loss in $1,000’s
Investment Choice (Action)
Large
Factory
Average
Factory
Small
Factory
Strong Economy (.3)
Stable Economy (.5)
Weak Economy (.2)
0
70
140
110
0
50
160
90
0
Example:
EOL (Large factory) =
0(.3) +
70(.5) +
(140)(.2) = 63
Expected Op. Loss (EOL) 63 43 93
Minimize expected op. loss by
choosing Average factory
Opportunity Loss Table
Goal: Minimize expected opportunity loss

21. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-21
Expected Opportunity Loss
vs. Expected Monetary Value
 The Expected Monetary Value (EMV) and the
Expected Opportunity Loss (EOL) criteria are
equivalent.
 Note that in this example the expected monetary
value solution and the expected opportunity loss
solution both led to the choice of the average size
factory.

22. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-22
Value of Information
 Expected Value of Perfect Information, EVPI
Expected Value of Perfect Information
EVPI = Expected profit under certainty
– expected monetary value of the best alternative
(EVPI is equal to the expected opportunity loss
from the best decision)

23. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-23
Expected Profit Under Certainty
 Expected profit under certainty
= expected value of the best decision, given perfect information
States of Nature
(Events)
Profit in $1,000’s
Investment Choice (Action)
Large
Factory
Average
Factory
Small
Factory
Strong Economy (.3)
Stable Economy (.5)
Weak Economy (.2)
200
50
-120
90
120
-30
40
30
20
Example: Best
decision given
“Strong Economy”
is “Large factory”
200
120
20
Value of best decision
for each event:

24. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-24
Expected Profit Under Certainty
States of Nature
(Events)
Profit in $1,000’s
Investment Choice (Action)
Large
Factory
Average
Factory
Small
Factory
Strong Economy (.3)
Stable Economy (.5)
Weak Economy (.2)
200
50
-120
90
120
-30
40
30
20
200
120
20
(continued)
Now weight these
outcomes with
their probabilities
to find the
expected value.
200(.3)+120(.5)+20(.2) = 124
Expected
profit under
certainty

25. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-25
Value of Information Solution
Expected Value of Perfect Information (EVPI)
EVPI = Expected profit under certainty
– Expected monetary value of the best decision
so: EVPI = 124 – 81
= 43
Recall: Expected profit under certainty = 124
EMV is maximized by choosing “Average factory”,
where EMV = 81
(EVPI is the maximum you would be willing to spend to obtain
perfect information)

26. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-26
Accounting for Variability
States of Nature
(Events)
Percent Return
Stock Choice (Action)
Stock A Stock B
Strong Economy (.7) 30 14
Weak Economy (.3) -10 8
Consider the choice of Stock A vs. Stock B
Expected Return: 18.0 12.2
Stock A has a higher
EMV, but what about
risk?

27. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-27
States of Nature
(Events)
Percent Return
Stock Choice
(Action)
Stock A Stock B
Strong Economy (.7) 30 14
Weak Economy (.3) -10 8
Variance: 336.0 7.56
Calculate the variance and standard deviation for
Stock A and Stock B:
Expected Return: 18.0 12.2
0.336)3(.)1810()7(.)1830(
)(μ)(σ
22
1
22
A

 
N
i
ii XPX
Example
Standard Deviation 18.33 2.75
Accounting for Variability
(continued)

28. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-28
Calculate the coefficient of variation for each stock:
%83.101100%
0.18
33.18
100%
EMV
σ
CV
A
A
A 
(continued)
%54.22100%
2.12
75.2
100%
EMV
σ
CV
B
B
B 
Stock A has
much more
relative
variability
Accounting for Variability

29. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-29
Return-to-Risk Ratio
Return-to-Risk Ratio (RTRR):
jσ
EMV(j)
RTRR(j) 
Expresses the relationship between the return
(expected payoff) and the risk (standard deviation)

30. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-30
Return-to-Risk Ratio
jσ
EMV(j)
RTRR(j) 
You might want to consider Stock B if you don’t
like risk. Although Stock A has a higher Expected
Return, Stock B has a much larger return to risk
ratio and a much smaller CV
982.0
33.18
0.18
σ
EMV(A)
RTRR(A)
A

436.4
75.2
2.12
σ
EMV(B)
RTRR(B)
B


31. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-31
Decision Making
with Sample Information
 Permits revising old
probabilities based on new
information
New
Information
Revised
Probability
Prior
Probability

32. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-32
Revised Probabilities
Example
Additional Information: Economic forecast is strong economy
 When the economy was strong, the forecaster was correct
90% of the time.
 When the economy was weak, the forecaster was correct 70%
of the time.
Prior probabilities
from stock choice
example
F1 = strong forecast
F2 = weak forecast
E1 = strong economy = 0.70
E2 = weak economy = 0.30
P(F1 | E1) = 0.90 P(F1 | E2) = 0.30

33. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-33
Revised Probabilities
Example
 Revised Probabilities (Bayes’ Theorem)
3.)E|F(P,9.)E|F(P 2111 
3.)E(P,7.)E(P 21 
875.
)3)(.3(.)9)(.7(.
)9)(.7(.
)F(P
)E|F(P)E(P
)F|E(P
1
111
11 


125.
)F(P
)E|F(P)E(P
)F|E(P
1
212
12 
(continued)

34. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-34
EMV with
Revised Probabilities
EMV Stock A = 25.0
EMV Stock B = 13.25
Revised
probabilities
Pi Event Stock A xijPi Stock B xijPi
.875 strong 30 26.25 14 12.25
.125 weak -10 -1.25 8 1.00
Σ = 25.0 Σ = 13.25
Maximum
EMV

35. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-35
EOL Table with
Revised Probabilities
EOL Stock A = 2.25
EOL Stock B = 14.00
Revised
probabilities
Pi Event Stock A xijPi Stock B xijPi
.875 strong 0 0 16 14.00
.125 weak 18 2.25 0 0
Σ = 2.25 Σ = 14.00
Minimum
EOL

36. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-36
States of Nature
(Events)
Percent Return
Stock Choice (Action)
Stock A Stock B
Strong Economy (.875) 30 14
Weak Economy (.125) -10 8
Variance: 175.00 3.94
Calculate the variance and standard deviation for
Stock A and Stock B:
Expected Return: 25.00 13.25
0.175)125(.)2510()875(.)2530(
)X(Pμ)X(σ
22
N
1i
i
2
i
2
A

 
Example
Standard Deviation: 13.23 1.98
Accounting for Variability with
Revised Probabilities

37. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-37
The coefficient of variation for each stock using the
results from the revised probabilities:
%92.52100%
0.25
229.13
100%
EMV
σ
CV
A
A
A 
(continued)
%97.14100%
25.13
984.1
100%
EMV
σ
CV
B
B
B 
Accounting for Variability with
Revised Probabilities

38. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-38
Return-to-Risk Ratio with
Revised Probabilities
With the revised probabilities, both stocks have
higher expected returns, lower CV’s, and larger
return to risk ratios
890.1
229.13
0.25
σ
EMV(A)
RTRR(A)
A

677.6
984.1
25.13
σ
EMV(B)
RTRR(B)
B


39. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-39
Utility
 Utility is the pleasure or satisfaction
obtained from an action.
 The utility of an outcome may not be the same
for each individual.

40. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-40
Utility
 Example: each incremental $1 of profit does not
have the same value to every individual:
 A risk averse person, once reaching a goal,
assigns less utility to each incremental $1.
 A risk seeker assigns more utility to each
incremental $1.
 A risk neutral person assigns the same utility to
each extra $1.

41. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-41
Three Types of Utility Curves
$ $ $
Risk Averter Risk Seeker Risk-Neutral

42. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-42
Maximizing Expected
Utility
 Making decisions in terms of utility, not $
 Translate $ outcomes into utility outcomes
 Calculate expected utilities for each action
 Choose the action to maximize expected utility

43. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 17-43
Chapter Summary
 Described the payoff table and decision trees
 Opportunity loss
 Provided criteria for decision making
 Maximax and Maximin
 Expected monetary value
 Expected opportunity loss
 Return to risk ratio
 Introduced expected profit under certainty and the value of
perfect information
 Discussed decision making with sample information
 Addressed the concept of utility