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

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}

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

-120

90

120

-30

40

30

20

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

Opportunity Loss

Profit in $1,000’s

(Events)

Investment Choice

(Action)

Large

Factory

Average

Factory

Small

Factory

Strong Economy

Stable Economy

Weak Economy

200

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 optimal payoff, given a particular event

Opportunity Loss

Opportunity Loss in

$1,000’s

(Events)

Investment Choice (Action)

Large

Factory

Average

Factory

_Factory

Small

Strong Economy

Stable Economy

Weak Economy

0

Payoff

Table

Opportunity

Loss Table

Profit in

$1,000’s

(Events)

Investment Choice (Action)

Large

Factory

Average

Factory

Small

Factory

Strong Economy

Stable Economy

Weak Economy

Decision Criteria



_{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}

Expected Monetary Value



_{The expected monetary value is the weighted}

average payoff, given specified probabilities for

each event







N

i

i

ij

P

X

j

EMV

1

)

(

Where EMV(j) = expected monetary value of action j

X

_ij

= payoff for action j when event i occurs

P

_i

= probability of event i

Expected Monetary Value



_{The expected value is the weighted average}

payoff, given specified probabilities for

each event

Suppose these

probabilities

have been

assessed for

these three

events

Profit in $1,000’s

(Events)

Investment Choice

(Action)

Large

Factory

Average

Factory

Small Factory

Strong Economy (0.3)

Stable Economy (0.5)

Weak Economy (0.2)

Expected Monetary Value



_{Example: EMV (Average factory) = 90(.3) + 120(.5)}

+ (-30)(.2) = 81

Payoff Table:

Profit in $1,000’s

(Events)

Investment Choice

(Action)

Large

Factory

Average

Factory

Factory

Small

Strong Economy (0.3)

Stable Economy (0.5)

Weak Economy (0.2)

200

50

-120

90

120

-30

Expected Opportunity Loss



_{The expected opportunity loss is the weighted}

average loss, given specified probabilities for each

event







N

1

i

i

ij

P

L

EOL(j)

Where EOL(j) = expected monetary value of action j

L

_ij

= opportunity loss for action j when event i occurs

P

_i

= probability of event i

Expected Opportunity Loss



_{Example: EOL (Large factory) = 0(.3) + 70(.5) +}

(140)(.2) = 63

Opportunity Loss Table

Opportunity Loss in

$1,000’s

(Events)

Investment Choice (Action)

Large

Factory

Average

Factory

_Factory

Small

Strong Economy (0.3)

Stable Economy (0.5)

Weak Economy (0.2)

0

70

140

110

0

50

160

90

0

Expected Opportunity

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}

Expected Profit Under Certainty



_Expected

profit under

certainty

= expected

value of the

best decision,

given perfect

information

Example: Best decision given “Strong

Economy” is “Large factory”

200 120 20

Value of best decision

for each event:

Profit in $1,000’s

(Events)

Investment Choice

(Action)

Large

Factory

Average

Factory

Small Factory

Strong Economy (0.3)

Stable Economy (0.5)

Weak Economy (0.2)

Expected Profit Under Certainty

200 120 20

Profit in $1,000’s

(Events)

Investment Choice

(Action)

Large

Factory

Average

Factory

Small Factory

Strong Economy (0.3)

Stable Economy (0.5)

Weak Economy (0.2)

200

these outcomes

with their

probabilities to

find the

expected profit

under

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

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

18.0

12.2

Consider the choice of Stock A vs. Stock B

Accounting for Variability

Calculate the variance and standard deviation

0

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

Variance

336.0

7.56

Accounting for Variability

Calculate the coefficient of variation for each stock:

%

relative

Return-to-Risk Ratio

Return-to-Risk Ratio (RTRR):

j

σ

EMV(j)

RTRR(j)

_

Return-to-Risk Ratio

j

σ

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.

Decision Making

with Sample Information



_{Permits revising old}

probabilities based on new

information

New

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.

Prior probabilities

from stock choice

example

F

₁

= strong forecast

F

₂

= weak forecast

E

₁

= strong economy = 0.70

E

₂

= weak economy = 0.30

Revised Probabilities

Example

Revised Probabilities (Bayes’ Theorem)

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

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

0

0

16

14.00

.125

weak

18

2.25

0

0

Σ = 2.25 Σ = 14.00

Accounting for Variability with

Revised Probabilities

Calculate the variance and standard deviation

0

Percent Return

(Events)

Stock Choice

(Action)

Stock A

Stock B

Strong Economy (.875)

30

14

Weak Economy (.125)

-10

8

Expected Return (EMV)

25.0

13.25

Variance

175.0

3.94

Accounting for Variability with

Revised Probabilities

The coefficient of variation for each stock using the

results from the revised probabilities:

Return-to-Risk Ratio with

Revised Probabilities

Utility



_{Utility is the pleasure or satisfaction obtained}

from an action.



_{The utility of an outcome may not be the}

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}

Three Types of Utility Curves

U

ti

li

ty

$

$

$

U

ti

li

ty

U

ti

li

ty

Maximizing Expected Utility



_{Making decisions in terms of utility, not $}



_{Translate $ outcomes into utility outcomes}



_{Calculate expected utilities for each action}

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}