Basic Business Statistics
(9th Edition)
Chapter 17
Decision Making
© 2004 Prentice-Hall, Inc.
Chap 17-1
Chapter Topics

The Payoff Table and Decision Trees


Criteria for Decision Making






Opportunity loss
Expected monetary value
Expected opportunity loss
Return-to-risk ratio
Expected Profit Under Certainty
Decision Making with Sample Information
Utility
© 2004 Prentice-Hall, Inc.
Chap 17-2
Features of Decision Making



List Alternative Courses of Action
List Possible Events or Outcomes or States of
Nature
Determine “Payoffs”


Associate a payoff with each course of action and
each event pair
Adopt Decision Criteria

Evaluate criteria for selecting the best course of
action
© 2004 Prentice-Hall, Inc.
Chap 17-3
List Possible Actions or Events
Two Methods
of Listing
Payoff Table
© 2004 Prentice-Hall, Inc.
Decision Tree
Chap 17-4
Payoff Table (Step 1)
Consider a food vendor determining
whether to sell soft drinks or hot dogs.
Course of Action (Aj)
Sell Soft Drinks (A1) Sell Hot Dogs (A2)
Event (Ei)
Cool Weather (E1)
x11 =$50
x12 = $100
Warm Weather (E2)
x21 = $200
x22 = $125
xij = payoff (profit) for event i and action j
© 2004 Prentice-Hall, Inc.
Chap 17-5
Payoff Table (Step 2)
Do Some Actions Dominate?




Action A “dominates” action B if the payoff of
action A is at least as high as that of action B
under any event and is higher under at least
one event
Action A is “inadmissible” if it is dominated by
any other action(s)
Inadmissible actions do not need to be
considered
Non-dominated actions are called “admissible”
© 2004 Prentice-Hall, Inc.
Chap 17-6
Payoff Table (Step 2)
Do Some Actions Dominate?
(continued)
Event (Ei)
Level of Demand
Low
Moderate
High


© 2004 Prentice-Hall, Inc.
Course of Action (Aj)
Production Process
A
B
C
70
80
100
120
120
125
200
180
160
D
100
120
150
Action C “dominates” Action D
Action D is “inadmissible”
Chap 17-7
Decision Tree:
Example
Food Vendor Profit Tree Diagram
x11 = $50
x21 = $200
x12 = $100
x22 =$125
© 2004 Prentice-Hall, Inc.
Chap 17-8
Opportunity Loss:
Example
Highest possible profit for an event Ei
- Actual profit obtained for an action Aj
Opportunity Loss (lij )
Event: Cool Weather
Action: Soft Drinks
Profit x11 : $50
Alternative Action: Hot Dogs
Profit x12 : $100
Opportunity Loss l11 = $100 - $50 = $50
Opportunity Loss l12 = $100 - $100 = $0
© 2004 Prentice-Hall, Inc.
Chap 17-9
Opportunity Loss: Table
Alternative Course of Action
Event
Optimal
Action
Profit of
Optimal
Action
Cool
Weather
Hot
Dogs
100
100 - 50 = 50
100 - 100 = 0
Warm
Weather
Soft
Drinks
200
200 - 200 = 0
200 - 125 = 75
© 2004 Prentice-Hall, Inc.
Sell Soft Drinks Sell Hot Dogs
Chap 17-10
Decision Criteria

Expected Monetary Value (EMV)


Expected Opportunity Loss (EOL)


Expected loss for taking action Aj
Expected Value of Perfect Information (EVPI)


Expected profit for taking action Aj
Expected opportunity loss from the best decision
Return-to-Risk-Ratio

Expected monetary value relative to the amount of
risk (variation)
© 2004 Prentice-Hall, Inc.
Chap 17-11
Decision Criteria - EMV
Expected Monetary Value (EMV) =
Sum (monetary payoffs of events)  (probabilities of the events)
Number of events
N
Vj 
 Xij Pi
i=1
EMVj = expected monetary value of action j
Xij = payoff for action j and event i
Pi = probability of event i occurring
© 2004 Prentice-Hall, Inc.
Chap 17-12
Decision Criteria - EMV Table
Example: Food Vendor
Pi Event
.50 Cool
.50 Warm
MV
xijPi
Soft
Drinks
$50
$50 .5 = $25
$200
$200 .5 = 100
EMV Soft Drink = $125
MV
Hot
Dogs
$100
xijPi
$100.50 = $50
$125
$125.50 = 62.50
EMV Hot Dog = $112.50
Highest EMV = Better Alternative
© 2004 Prentice-Hall, Inc.
Chap 17-13
Decision Criteria - EOL
Expected Opportunity Loss (EOL) =
Sum
(opportunity losses of events)  (probabilities of events)
Lj 
N
 lij Pi
i =1
© 2004 Prentice-Hall, Inc.
EOLj = expected opportunity loss of action j
lij = opportunity loss for action j and event i
Pi = probability of event i occurring
Chap 17-14
Decision Criteria - EOL Table
Example: Food Vendor
Pi
Event Op Loss
Soft Drinks
.50 Cool
.50 Warm
lijPi
Op Loss
Hot Dogs
$50
$50.50 = $25
$0
0
$0 .50 = $0
$75
EOL Soft Drinks = $25
lijPi
$0.50 = $0
$75 .50 = $37.50
EOL Hot Dogs = $37.50
Lowest EOL = Better Choice
© 2004 Prentice-Hall, Inc.
Chap 17-15
EVPI

Expected Value of Perfect Information (EVPI)

The expected opportunity loss from the best
decision
Expected Profit Under Certainty
-
Expected Monetary Value of the Best Alternative
EVPI (should be a positive number)

Represents the maximum amount you are
willing to pay to obtain perfect information
© 2004 Prentice-Hall, Inc.
Chap 17-16
EVPI Computation
Expected Profit Under Certainty
= .50($100) + .50($200)
= $150
Expected Monetary Value of the Best Alternative
= $125
EVPI = $150 - $125 = $25
= Lowest EOL
= The maximum you would be willing to
spend to obtain perfect information
© 2004 Prentice-Hall, Inc.
Chap 17-17
Taking Account of Variability
Example: Food Vendor
2 for Soft Drink
= (50 -125)2 .5 + (200 -125)2 .5 = 5625
 for Soft Drink = 75
CVfor Soft Drinks = (75/125)  100% = 60%
2 for Hot Dogs = 156.25  for Hot Dogs = 12.5
CVfor Hot Dogs = (12.5/112.5)  100% = 11.11%
© 2004 Prentice-Hall, Inc.
Chap 17-18
Return-to-Risk Ratio

Expresses the Relationship between the
Return (Expected Payoff) and the Risk
(Standard Deviation)


RTRR = Return-to-Risk Ratio =
EMV j
j
1
RTRR = Return-to-Risk Ratio =
CV j
© 2004 Prentice-Hall, Inc.
Chap 17-19
Return-to-Risk Ratio
Example: Food Vendor
RTRR Soft Drinks = 1/CVSoft Drinks = 1.67
RTRR Hot Dogs = 1/CVHot Dogs = 9
You might wish to choose Hot Dogs. Although
Soft Drinks have the higher Expected Monetary
Value, Hot Dogs have a much larger return-torisk ratio and a much smaller CV.
© 2004 Prentice-Hall, Inc.
Chap 17-20
Decision Making in PHStat

PHStat | Decision Making | Expected Monetary
Value


Check the “Expected Opportunity Loss” and “Measures
of Variation” boxes
Excel Spreadsheet for the Food Vendor Example
© 2004 Prentice-Hall, Inc.
Chap 17-21
Decision Making with Sample
Information
Prior
Probability

Permits Revising Old
Probabilities Based on
New Information
New
Information
Revised
Probability
© 2004 Prentice-Hall, Inc.
Chap 17-22
Revised Probabilities
Example: Food Vendor
Additional Information: Weather forecast is COOL.
When the weather has been cool, the forecaster has been
correct 80% of the time.
When it has been warm, the forecaster has been correct
70% of the time.
F1 = Cool forecast
F2 = Warm forecast
E1 = Cool weather = 0.50
Prior
Probability
E2 = Warm weather = 0.50
P(F1 | E1) = 0.80 P(F1 | E2) = 0.30
© 2004 Prentice-Hall, Inc.
Chap 17-23
Revising Probabilities
Example: Food Vendor

Revised Probability (Bayes’ Theorem)
P  F1 | E1   0.80 P  F1 | E2   0.30
P  E1   0.50 P  E2   0.50
P  E1  P  F1 | E1 
.50 .80 

P  E1 | F1  

 .73
P  F1 
.50 .80   .50 .30 
P  E2  P  F1 | E2 
P  E2 | F1  
 .27
P  F1 
© 2004 Prentice-Hall, Inc.
Chap 17-24
Revised EMV Table
Example: Food Vendor
Pi
Event
.73 Cool
Soft
Drinks
$50
.27 Warm
$200
xijPi
$36.50
54
EMV Soft Drink = $90.50
Hot
Dogs
$100
125
xijPi
$73
33.75
EMV Hot Dog = $106.75
Revised Probabilities
Highest EMV = Better Alternative
© 2004 Prentice-Hall, Inc.
Chap 17-25
Revised EOL Table
Example: Food Vendor
Pi
Event Op Loss
Soft Drink
.73 Cool
.27 Warm
$50
0
lijPi
$36.50
$0
EOL Soft Drinks = 36.50
OP Loss
Hot Dogs
lijPi
$0
0
75
20.25
EOL Hot Dogs = $20.25
Lowest EOL = Better Choice
© 2004 Prentice-Hall, Inc.
Chap 17-26
Revised EVPI Computation
Expected Profit Under Certainty
= .73($100) + .27($200)
= $127
Expected Monetary Value of the Best Alternative
= $106.75
EPVI = $127 - $106.75 = $20.25
= The maximum you would be willing to
spend to obtain perfect information
© 2004 Prentice-Hall, Inc.
Chap 17-27
Taking Account of Variability:
Revised Computation
2 for Soft Drinks
= (50 -90.5)2 .73 + (200 -90.5)2 .27 = 4434.75
 for Soft Drinks = 66.59
CVfor Soft Drinks = (66.59/90.5)  100% = 73.6%
2 for Hot Dogs = 123.1875
 for Hot Dogs = 11.10
CVfor Hot Dogs = (11.10/106.75)  100% = 10.4%
© 2004 Prentice-Hall, Inc.
Chap 17-28
Revised Return-to-Risk Ratio
RTRR Soft Drinks = 1/CVSoft Drinks = 1.36
RTRR Hot Dogs = 1/CVHot Dogs = 9.62
You might wish to choose Hot Dogs. Hot Dogs
have a much larger return-to-risk ratio.
© 2004 Prentice-Hall, Inc.
Chap 17-29
Revised Decision Making
in PHStat

PHStat | Decision Making | Expected Monetary
Value



Check the “Expected Opportunity Loss” and
“Measures of Variation” boxes
Use the revised probabilities
Excel Spreadsheet for the Food Vendor Example
© 2004 Prentice-Hall, Inc.
Chap 17-30
Utility

Utility is the Idea that Each Incremental $1 of
Profit Does Not Have the Same Value to Every
Individual



A risk averse person, once reaching a goal, assigns
less value to each incremental $1.
A risk seeker assigns more value to each
incremental $1.
A risk-neutral person assigns the same value to
each incremental $1.
© 2004 Prentice-Hall, Inc.
Chap 17-31
Three Types of Utility Curves
$
$
$
Risk Averter:
Risk Seeker:
Risk-Neutral:
Utility rises slower
than payoff
Utility rises faster
than payoff
Maximizes
expected payoff
and ignores risk
© 2004 Prentice-Hall, Inc.
Chap 17-32
Chapter Summary

Described the Payoff Table and Decision Trees


Provided Criteria for Decision Making






Opportunity loss
Expected monetary value
Expected opportunity loss
Return-to-risk ratio
Introduced Expected Profit Under Certainty
Discussed Decision Making with Sample
Information
Addressed the Concept of Utility
© 2004 Prentice-Hall, Inc.
Chap 17-33