Quantitative Decision Techniques

13/04/2009Decision Trees and Utility TheoryDecision Trees and Utility Theory

Chapter Outline

4.1 Introduction

4.2 Decision Trees

4.3 How Probability Values Are Estimated by

Bayesian Analysis

4.4 Utility Theory

4.5 Sensitivity Analysis

Introduction

Decision trees enable one to look at decisions:

• with many alternatives and states of nature

• which must be made in sequence

Decision Trees

A graphical representation where:

a decision node from which one of several alternatives may be chosen

a state-of-nature node out of which one state of nature will occur

Thompson’s Decision Tree Fig. 4.1

1

2

A Decision Node

A State of Nature Node

Favorable Market

Unfavorable Market

Favorable Market

Unfavorable Market

Construct

Large P

lant

Construct Small Plant

Do Nothing

Five Steps toDecision Tree Analysis

1. Define the problem2. Structure or draw the decision tree3. Assign probabilities to the states of nature4. Estimate payoffs for each possible

combination of alternatives and states of nature

5. Solve the problem by computing expected monetary values (EMVs) for each state of nature node.

Decision Table for Thompson Lumber

AlternativeState of Nature

Favorable Market ($)

Unfavorable Market ($)

Construct a large plant

200,000 -180,000

Construct a small plant

100,000 -20,000

Do nothing 0 0

Probabilities

0.50 0.50

Thompson’s Decision Tree Fig. 4.2

A Decision Node

A State of Nature Node

Favorable Market (0.5)

Unfavorable Market (0.5)

FavorableMarket (0.5)

Unfavorable Market (0.5)

Constru

ct Larg

e

Plant

Construct Small Plant

Do Nothing

$200,000

-$180,000

$100,000

-$20,000

0

EMV =$40,000

EMV=$10,000

1

2

2nd Decision Table for Thompson Lumber

AlternativeState of Nature

Favorable Results

Unfavorable Results

Favorable Market

0.78 0.27

Unfavorable Market

0.22 0.73

Probabilities

0.45 0.55

Thompson’s Decision Tree -Fig. 3

Thompson’s Decision Tree -Fig. 4

Expected Value of Sample Information

Expected value of best

decision with sample

information, assuming no

cost to gather it

Expected value of best

decision without

sample informationEVSI =

Expected Value of Sample Information

EVSI= EV of best decision with sample information, assuming

no cost to gather it– EV of best decision without sample information= EV with sample info. + cost – EV without sample info.DM could pay up to EVSI for a survey.If the cost of the survey is less than EVSI, it is indeed

worthwhile.

In the example:EVSI = $49,200 + $10,000 – $40,000 = $19,200

Estimating Probability Values by Bayesian Analysis

• Management experience or intuition

• History

• Existing data

• Need to be able to revise probabilities based upon new data

Posteriorprobabilities

Priorprobabilities New data

Bayes Theorem

Example:• Market research specialists have told DM that,

statistically, of all new products with a favorable market, market surveys were positive and predicted success correctly 70% of the time.

• 30% of the time the surveys falsely predicted negative result

• On the other hand, when there was actually an unfavorable market for a new product, 80% of the surveys correctly predicted the negative results.

• The surveys incorrectly predicted positive results the remaining 20% of the time.

Bayesian Analysis

Market Survey Reliability

Actual States of Nature

Result of Survey Favorable

Market (FM)

Unfavorable

Market (UM)

Positive (predicts

favorable market

for product)

P(survey positive|FM) = 0.70

P(survey positive|UM) = 0.20

Negative (predicts

unfavorable

market for

product)

P(survey negative|FM) = 0.30

P(survey negative|UM) = 0.80

Calculating Posterior Probabilities

P(BA) P(A)P(AB) =

P(BA) P(A) + P(BA’) P(A’)where A and B are any two events, A’ is the complement of A

P(FMsurvey positive) = [P(survey positiveFM)P(FM)] / [P(survey positiveFM)P(FM) + P(survey positiveUM)P(UM)]

P(UMsurvey positive) = [P(survey positiveUM)P(UM)] / [P(survey positiveFM)P(FM) + P(survey positiveUM)P(UM)]

Probability Revisions Given a Positive Survey

Conditional

ProbabilityPosterior

Probability

State

of

Nature

P(Survey positive|State of Nature

Prior

Probability

Joint

Probability

FM 0.70 * 0.50 0.350.450.35 = 0.78

UM 0.20 * 0.500.45

0.10 0.10 = 0.22

0.45 1.00

Probability Revisions Given a Negative Survey

Conditional

ProbabilityPosterior

Probability

State

of

Nature

P(Survey

negative|State

of Nature)

Prior Probability

Joint Probability

FM 0.30 * 0.50 0.150.55

0.15 = 0.27

UM 0.80 * 0.50 0.400.55

0.40 = 0.73

0.55 1.00

Utility Theory

• Utility assessment assigns the worst outcome a utility of 0, and the best outcome, a utility of 1.

• A standard gamble is used to determine utility values: When you are indifferent, the utility values are equal.

• Choose the alternative with the maximum expected utility EU(ai) = u(ai) = u(vij) P(j)

j

Utility Theory

$5,000,000

$0

$2,000,000

Accept Offer

Reject Offer

Red(0.5)

Blue(0.5)

Utility Assessment

• Utility assessment assigns the worst outcome a

utility of 0, and the best outcome, a utility of 1.

• A standard gamble is used to determine utility

values.

• When you are indifferent, the utility values are

equal.

Standard Gamble for Utility Assessment

Best outcomeUtility = 1

Worst outcomeUtility = 0

Other outcomeUtility = ??

(p)

(1-p)Alternative 1

Alternative 2

Figure 4.7$10,000U($10,000) = 1.0

0U(0)=0

$5,000U($5,000)=p=0.80

p= 0.80

(1-p)= 0.20Invest in

Real Estate

Invest in Bank

Utility Assessment (1st approach)

v*u(v*) = 1

x1u(x1) = 0.5

x2u(x2) = 0.75

(0.5)

(0.5)Lottery ticket

Certain money

Best outcome (v*)u(v*) = 1

Worst outcome (v–)u(v–) = 0

Certain outcome (x1)u(x1) = 0.5

(0.5)

(0.5)Lottery ticket

Certain money

x1u(v*) = 0.5

Worst outcome (v–)u(v–) = 0

x3u(x3) = 0.25

(0.5)

(0.5)Lottery ticket

Certain money

In the example:u(-180) = 0 and u(200) = 1X1= 100 u(100) = 0.5X2 = 175 u(175) = 0.75X3 = 5 u(5) = 0.25

I II

III

0

0.2

0.4

0.6

0.8

1

-200 -150 -100 -50 0 50 100 150 200

Utility Assessment (2nd approach)

Best outcome (v*)u(v*) = 1

Worst outcome (v–)u(v–) = 0

Certain outcome (vij)u(vij) = p

(p)

(1–p)Lottery ticket

Certain money

In the example:u(-180) = 0 and u(200) = 1

For vij=–20, p=%70 u(–20) = 0.7

For vij=0, p=%75 u(0) = 0.75

For vij=100, p=%90 u(100) = 0.9

0

0.2

0.4

0.6

0.8

1

-200 -150 -100 -50 0 50 100 150 200

Utilities STATES OF NATURE

ALTERNATIVESFavorable

marketUnfavorable

marketExpected

UtilityConstruct large plant 1 0 0.6Construct small plant 0.9 0.7 0.82Do nothing 0.75 0.75 0.75PROBABILITIES 0.6 0.4

Sample Utility Curve

00.10.20.30.40.50.60.70.80.91

$- $2,000 $4,000 $6,000 $8,000 $10,000

Monetary Value

Util

ity

Preferences for Risk

Monetary Outcome

RiskA

void

er

RiskSe

ekerRisk

Indi

ffere

nce

Uti

lity

Example

Point up (0.45)

Point down (0.55)

$10,000

-$10,000

0

Alternative 1

Play the game

Alternative 2Do not play the game

Utility Curve for Example

0

0.1

0.20.3

0.4

0.5

0.6

0.70.8

0.9

1

-$20,000 -$10,000 $0 $10,000 $20,000 $30,000

Using Expected Utilities in Decision Making

Tack landspoint up (0.45)

Tack lands point down (0.55)

0.30

0.05

0.15

Alternative 1

Play the game

Alternative 2Don’t play

Utility