Chapter17 - Decision Making
Statistics for Managers Using Microsoft® Excel 5th Edition
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
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 Event/Outcome combination
Adopt Decision Criteria
Evaluate Criteria for Selecting the Best Course of Action Payoff Table Profit in $1,000’s
(Events) Investment Choice (Action)
Large Factory Average Factory
Small Factory Strong Economy Stable Economy Weak Economy 200
50
90 120
40
30
- 120
- 30
20 A payoff table shows alternatives, states of nature, and payoffs Decision Tree Strong Economy
200 Large factory
Stable Economy
50 Weak Economy -120 Strong Economy
90 Average factory Stable Economy
120 Weak Economy
- -30
Strong Economy
40 Small factory Stable Economy
30 Weak Economy
20 Payoffs Opportunity loss is the difference between an actual payoff for an action and the optimal payoff, given a particular event Payoff Table
Opportunity Loss Profit in $1,000’s
(Events)
Investment Choice
(Action)
Large Factory
Average Factory
Small Factory
Strong Economy Stable Economy Weak Economy
200
50
90 120
40
30
- 120
- 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.
20
Profit in $1,000’s
30
40
90 120
50
200
Strong Economy Stable Economy Weak Economy
Small Factory
Average Factory
Large Factory
Investment Choice (Action)
(Events)
Table Opportunity Loss Table
- 30
90 Payoff
50 160
110
70 140
Strong Economy Stable Economy Weak Economy
Small Factory
Average Factory
Large Factory
Investment Choice (Action)
(Events)
Opportunity Loss Opportunity Loss in $1,000’s
- 120
Expected Monetary Value (EMV)
The expected profit for taking action A j
Expected Opportunity Loss (EOL)
The expected opportunity loss for taking action A j
Expected Value of Perfect Information (EVPI)
The expected opportunity loss from the best decision
Expected Monetary Value Goal: Maximize expected value
The expected monetary value is the weighted
average payoff, given specified probabilities for each event N
EMV ( j )
X P ij i
i
1
Where EMV(j) = expected monetary value of action j X = payoff for action j when event i occurs ij P = probability of event i i
Expected Monetary Value
The expected value is the weighted average payoff, given specified probabilities for each event Investment Choice
Profit in $1,000’s (Action) Suppose these
(Events)
probabilities
Large Average Small Factory Factory Factory
have been assessed for these three
Strong Economy (0.3) 200
90
40 Stable Economy (0.5) 50 120 30 events Weak Economy (0.2) -120 -30
20
31
200
81
61
20 EMV (Expected Values)
30
40
90 120
50
Factory Strong Economy (0.3) Stable Economy (0.5) Weak Economy (0.2)
Factory Small
Factory Average
(Action) Large
Investment Choice
(Events)
Payoff Table: Profit in $1,000’s
- 30
- (-30)(.2) = 81
Example: EMV (Average factory) = 90(.3) + 120(.5)
Expected Monetary Value
- 120
Expected Opportunity Loss Goal: Minimize expected opportunity loss
The expected opportunity loss is the weighted
average loss, given specified probabilities for each
eventN
EOL(j) L P
ij i
i
1
Where EOL(j) = expected monetary value of action j L = opportunity loss for action j when event i occurs ij P = probability of event i i
Expected Opportunity Loss Opportunity Loss Table
Investment Choice (Action) Opportunity Loss in
Large Average Small
$1,000’s
Factory Factory Factory
(Events) Strong Economy (0.3) 110 160 Stable Economy (0.5)
70
90 Weak Economy (0.2) 140
50 Expected Opportunity
63
43
93 Loss (EOL)
Example: EOL (Large factory) = 0(.3) + 70(.5) +
(140)(.2) = 63
Value of Information
Expected Value of Perfect Information EVPI = Expected profit under certainty
Expected Value of Perfect Information, EVPI
- – expected monetary value of the best alternative
EVPI is equal to the expected opportunity loss from the best decision
Expected Profit Under Certainty Investment Choice
Profit in $1,000’s (Action) profit under
Expected
(Events) Large Average Small Factory
certainty
Factory Factory
= expected value of the
Strong Economy (0.3) 200
90
40 Stable Economy (0.5) 50 120
30
best decision,
Weak Economy (0.2) -120 -30
20 given perfect information
Value of best decision 200 120 20 for each event: Example: Best decision given “Strong
Economy” is “Large factory” Expected Profit Under Certainty Investment Choice
Profit in $1,000’s (Action)
(Events) Large Average Small Factory
Factory Factory Strong Economy (0.3) 200
90
40 Stable Economy (0.5) 50 120
30 Now weight
Weak Economy (0.2) -120 -30
20
these outcomes with their probabilities to
200 120 20 find the expected profit
200(.3)+120(.5)+20(.2) under certainty:
= 124 Value of Information Solution
Expected Value of Perfect Information (EVPI) EVPI = Expected profit under certainty
- – Expected monetary value of the best decision
so:
Recall: Expected profit under certainty = 124 EMV is maximized by choosing “Average factory,” where EMV = 81
EVPI = 124 – 81 = 43
(EVPI is the maximum you would be willing to spend to obtain perfect information)
Accounting for Variability Percent Return
(Events)
Stock Choice
(Action) Stock A Stock B
Strong Economy (.7)
30
14 Weak Economy (.3)
- 10
8 Expected Return (EMV)
18.0
12.2 Consider the choice of Stock A vs. Stock B
Stock A has a higher EMV, but what about risk? Accounting for Variability Calculate the variance and standard deviation
Stock Choice
(Action)
Percent Return
Stock A Stock B (Events)
30
14 Strong Economy (.7)
- 10
8 Weak Economy (.3)
18.0
12.2 Expected Return (EMV) 336.0
7.56 Variance 2 N 2 2 2
18.33
2.75 Standard Deviation
Example: σ ( X μ) P ( X ) (
30
18 ) (. 7 ) (10 18 ) (. 3 ) 336 . A i i i 1 Accounting for Variability Calculate the coefficient of variation for each stock:
σ 18 .
Stock A has CV 100% 100% 101 . 83 % A EMV 18 . A much more relative variability
σ B 2 .
75 CV 100% 100% 22 . 54 % B EMV B 12 .
2 Return-to-Risk Ratio Return-to-Risk Ratio (RTRR):
EMV(j) RTRR(j)
σ
jExpresses the relationship between the return (expected payoff) and the risk (standard deviation)
Return-to-Risk Ratio EMV(j) RTRR(j)
σ j EMV(A)
18 .
. 982 σ A 18 .
33 EMV(B)
12 .
2 RTRR(B) 4 . 436 σ B 2 .
75 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.
Decision Making with Sample Information Prior Probability
New probabilities based on new
Permits revising old
Information information
Revised Probability 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.
F = strong forecast 1 F = weak forecast 2 Prior probabilities from stock choice E = strong economy = 0.70 1 example E = weak economy = 0.30 2 P(F | E ) = 0.90 P(F | E ) = 0.30 1 1 1 2
Revised Probabilities Example Revised Probabilities (Bayes’ Theorem)
1
2
1
1
2
2
1
) ( ) ( ) | ( ) | (
F P E P E F P F E P 125 .
1
1
1
) 3 . E | F ( P , ) 9 . E | F ( P
1
1
) ( ) ( ) | ( ) | (
) 7 )(. 9 (.
) 3 )(. ) 3 (. 7 )(. 9 (.
1 875 .
2
) 7 . E ( P
1 ) 3 . E ( P ,
1
1
2
F P E P E F P F E P EMV with Revised Probabilities EMV Stock A = 25.0 EMV Stock B = 13.25
Revised probabilities P i Event Stock A
X ij P i Stock B
X ij P i .875 strong
30
26.25
14
12.25 .125 weak -10 -1.25
8
1.00 Σ = 25.0 Σ = 13.25
Maximum EMV EOL Table with Revised Probabilities EOL Stock A = 2.25 EOL Stock B = 14.00
Revised probabilities P i Event Stock A
X ij P i Stock B
X ij P i .875 strong
16
14.00 .125 weak
18
2.25 Σ = 2.25 Σ = 14.00
Minimum EOL Accounting for Variability with Revised Probabilities Calculate the variance and standard deviation
Stock Choice
(Action)
Percent Return
Stock A Stock B (Events)
30
14 Strong Economy (.875)
- 10
8 Weak Economy (.125)
25.0
13.25 Expected Return (EMV) 175.0
3.94 Variance 2 N 2 2 2 13.229 1.984
Example:
Standard Deviation σ (
X μ) P ( X ) (
30 25 ) (. 875 ) (
10 25 ) (. 125 ) 175 . A i i i 1
Accounting for Variability with
Revised Probabilities The coefficient of variation for each stock using the results from the revised probabilities:% 92 . 52 100% .
25 229 .
13 100% EMV σ
CV A A A
% 97 . 14 100% 25 .
13 984 .
1 100% EMV σ
CV B B B
Return-to-Risk Ratio with Revised Probabilities EMV(A) 25 .
1 . 890 σ 13 . 229
A EMV(B) 13 .
25 RTRR(B) 6 . 678
σ 1 . 984 B
With the revised probabilities, both stocks have higher expected returns, lower CV’s, and larger return to risk ratios
Utility
from an action. The utility of an outcome may not be the
Utility is the pleasure or satisfaction obtained
same for each individual.
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.
incremental $1.
A risk seeker assigns more utility to each
A risk neutral person assigns the same utility
to each extra $1.
Three Types of Utility Curves U ti li ty $
$ $ U ti li ty
U ti li ty Risk Averter Risk Seeker Risk-Neutral
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
Chapter Summary
Described the payoff table and decision trees
Opportunity loss
Provided criteria for decision making
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 In this chapter, we have