basic business statistics, 10e © 2006 prentice-hall, inc. chap 17-1 chapter 17 decision making...

Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc. Chap 17-1

Chapter 17

Decision Making

Basic Business Statistics10th 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


List Possible Actions or Events

Payoff Table Decision Tree

Two Methods of

Listing


A Payoff Table

A payoff table shows alternatives, states of nature, and payoffs

Investment Choice

(Action)

Profit in $1,000’s

(Events)

Strong Economy

Stable Economy

Weak Economy

Large factory

Average factory

Small factory

200

90

40

50

120

30

-120

-30

20


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


Opportunity Loss

Investment Choice

(Action)


(Events)

Strong Economy

Stable Economy

Weak Economy

Large factoryAverage factorySmall factory

2009040

50120 30

-120-30 20The 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

Payoff Table


Opportunity Loss

Investment Choice

(Action)


(States of Nature)

Strong Economy

Stable Economy

Weak Economy


2009040

50120 30

-120-30 20

(continued)

Investment Choice

(Action)

Opportunity Loss in $1,000’s

(Events)

Strong Economy

Stable Economy

Weak Economy


0110160

70 0

90

140500

Payoff Table

Opportunity Loss Table


Decision Criteria

Expected Monetary Value (EMV) The expected profit for taking action Aj

Expected Opportunity Loss (EOL) The expected opportunity loss for taking action Aj

Expected Value of Perfect Information (EVPI) The expected opportunity loss from the best decision


Expected Monetary Value Solution

The expected monetary value is the weighted average payoff, given specified probabilities for each event

N

1iiijPx)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


The expected value is the weighted average payoff, given specified probabilities for each event

Investment Choice

(Action)


(Events)

Strong Economy

(.3)

Stable Economy

(.5)

Weak Economy

(.2)


2009040

50120 30

-120-30 20

Suppose these probabilities have been assessed for these three events

(continued)



Investment Choice

(Action)


(Events)

Strong Economy

(.3)

Stable Economy

(.5)

Weak Economy

(.2)


2009040

50120 30

-120-30 20

Example: EMV (Average factory) = 90(.3) + 120(.5) + (-30)(.2)

= 81

Expected Values

(EMV)618131

Maximize expected value by choosing Average factory

(continued)

Payoff Table:

Goal: Maximize expected value



Decision Tree Analysis

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


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)


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


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


Expected Opportunity Loss Solution

The expected opportunity loss is the weighted average loss, given specified probabilities for each event

N

1iiijPL)j(EOL

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

Lij = opp. loss for action j when event i occurs

Pi = probability of event i

Goal: Minimize expected opportunity loss


Expected Opportunity Loss Solution

Investment Choice

(Action)

Opportunity Loss in $1,000’s

(Events)

Strong Economy

(.3)

Stable Economy

(.5)

Weak Economy

(.2)


0110160

70 0

90

140500

Example: EOL (Large factory) = 0(.3) + 70(.5) + (140)(.2)

= 63

Expected Op. Loss

(EOL)634393

Minimize expected

op. loss by choosing Average factory

Opportunity Loss Table

Goal: Minimize expected opportunity loss


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.


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)


Expected Profit Under Certainty

Expected profit under certainty

= expected value of the best decision, given perfect information

Investment Choice

(Action)


(Events)

Strong Economy

(.3)

Stable Economy

(.5)

Weak Economy

(.2)


2009040

50120 30

-120-30 20

Example: Best decision given “Strong Economy” is “Large factory”

200 120 20Value of best decision for each event:


Expected Profit Under Certainty

Investment Choice

(Action)


(Events)

Strong Economy

(.3)

Stable Economy

(.5)

Weak Economy

(.2)


2009040

50120 30

-120-30 20

200 120 20

(continued)

Now weight these outcomes with their probabilities to find the expected value: 200(.3)+120(.5)+20(.2)

= 124Expected profit under certainty


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)


Accounting for Variability

Stock Choice

(Action)

Percent Return

(Events)

Strong Economy

(.7)

Weak Economy

(.3)

Stock A 30 -10

Stock B 14 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?


Stock Choice

(Action)

Percent Return

(Events)

Strong Economy

(.7)

Weak Economy

(.3)

Stock A 30 -10

Stock B 14 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()X(Pμ)X(σ 22N

1ii

2i

2A

Example:

Standard Deviation:

18.33

2.75

Accounting for Variability(continued)


Calculate the coefficient of variation for each stock:

%83.101100%0.18

33.18 100%

EMV

σCV

A

AA

(continued)

%54.22100%2.12

75.2 100%

EMV

σCV

B

BB

Stock A has much more relative variability

Accounting for Variability


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)


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

982.033.18

0.18

σ

EMV(A)RTRR(A)

A

436.475.2

2.12

σ

EMV(B)RTRR(B)

B


Decision Making with Sample Information

Permits revising old probabilities based on new information

NewInformation

RevisedProbability

PriorProbability


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


EMV with Revised Probabilities

EMV Stock A = 25.0

EMV Stock B = 11.25

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 = 11.25

Maximum EMV


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


Stock Choice

(Action)

Percent Return

(Events)

Strong Economy

(.875)

Weak Economy

(.125)

Stock A 30 -10

Stock B 14 8

Variance:

175.0

3.94

Calculate the variance and standard deviation for Stock A and Stock B:

Expected Return:

25.0

13.25

0.175)125(.)2510()875(.)2530()X(Pμ)Xi(σ 22N

1ii

22A

Example:

Standard Deviation:

13.229

1.984

Accounting for Variability with Revised Probabilities


The coefficient of variation for each stock using the results from the revised probabilities:

%92.52100%0.25

229.13 100%

EMV

σCV

A

AA

(continued)

%97.14100%25.13

984.1 100%

EMV

σCV

B

BB

Accounting for Variability with Revised Probabilities


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

890.1229.13

0.25

σ

EMV(A)RTRR(A)

A

011.7984.1

25.13

σ

EMV(B)RTRR(B)

B


Utility

Utility is the pleasure or satisfaction obtained from an action.

The utility of an outcome may not be the 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.

A risk seeker assigns more utility to each incremental $1.

A risk neutral person assigns the same utility to each extra $1.


Three Types of Utility CurvesU

t ili

ty

$ $ $

Uti

lity

Ut i

lity

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

basic business statistics, 10e © 2006 prentice-hall, inc. chap 17-1 chapter 17 decision making...

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