mcgraw-hill/irwin © the mcgraw-hill companies, inc., 2003 12.1 table of contents chapter 12...

© The McGraw-Hill Companies, Inc., 200312.1McGraw-Hill/Irwin

Table of ContentsChapter 12 (Decision Analysis)

Decision Analysis Examples 12.2–12.3A Case Study: The Goferbroke Company Problem (Section 12.1) 12.4–12.8Decision Criteria (Section 12.2) 12.9–12.13Decision Trees (Section 12.3) 12.14–12.19Sensitivity Analysis with Decision Trees (Section 12.4) 12.20–12.24Checking Whether to Obtain More Information (Section 12.5) 12.25–12.27Using New Information to Update the Probabilities (Section 12.6) 12.28–12.35Decision Tree to Analyze a Sequence of Decisions (Section 12.7) 12.36–12.39Sensitivity Analysis with a Sequence of Decisions (Section 12.8) 12.40–12.47Using Utilities to Better Reflect the Values of Payoffs (Section 12.9) 12.48–12.64

Introduction to Decision Analysis (UW Lecture) 12.65–12.80These slides are based upon a lecture from the MBA core course in Management Science at the University of Washington (as taught by one of the authors).

Sequential Decisions and the Value of Information (UW Lecture) 12.81–12.91These slides are based upon a lecture from the MBA core course in Management Science at the University of Washington (as taught by one of the authors).

Risk Attitude and Utility Functions (UW Lecture) 12.92–12.102These slides are based upon a lecture from the MBA core course in Management Science at the University of Washington (as taught by one of the authors).


Decision Analysis

• Managers often must make decisions in environments that are fraught with uncertainty.

• Some Examples– A manufacturer introducing a new product into the marketplace

• What will be the reaction of potential customers?

• How much should be produced?

• Should the product be test-marketed?

• How much advertising is needed?

– A financial firm investing in securities

• Which are the market sectors and individual securities with the best prospects?

• Where is the economy headed?

• How about interest rates?

• How should these factors affect the investment decisions?


Decision Analysis

• Managers often must make decisions in environments that are fraught with uncertainty.

• Some Examples– A government contractor bidding on a new contract.

• What will be the actual costs of the project?

• Which other companies might be bidding?

• What are their likely bids?

– An agricultural firm selecting the mix of crops and livestock for the season.

• What will be the weather conditions?

• Where are prices headed?

• What will costs be?

– An oil company deciding whether to drill for oil in a particular location.

• How likely is there to be oil in that location?

• How much?

• How deep will they need to drill?

• Should geologists investigate the site further before drilling?


The Goferbroke Company Problem

• The Goferbroke Company develops oil wells in unproven territory.

• A consulting geologist has reported that there is a one-in-four chance of oil on a particular tract of land.

• Drilling for oil on this tract would require an investment of about $100,000.

• If the tract contains oil, it is estimated that the net revenue generated would be approximately $800,000.

• Another oil company has offered to purchase the tract of land for $90,000.

Question: Should Goferbroke drill for oil or sell the tract?


Prospective Profits

Profit

Status of Land Oil Dry

Alternative

Drill for oil $700,000 –$100,000

Sell the land 90,000 90,000

Chance of status 1 in 4 3 in 4


Decision Analysis Terminology

• The decision maker is the individual or group responsible for making the decision.

• The alternatives are the options for the decision to be made.

• The outcome is affected by random factors outside the control of the decision maker. These random factors determine the situation that will be found when the decision is executed. Each of these possible situations is referred to as a possible state of nature.

• The decision maker generally will have some information about the relative likelihood of the possible states of nature. These are referred to as the prior probabilities.

• Each combination of a decision alternative and a state of nature results in some outcome. The payoff is a quantitative measure of the value to the decision maker of the outcome. It is often the monetary value.


Prior Probabilities

State of Nature Prior Probability

The tract of land contains oil 0.25

The tract of land is dry (no oil) 0.75


Payoff Table (Profit in $Thousands)

State of Nature

Alternative Oil Dry

Drill for oil 700 –100

Sell the land 90 90

Prior probability 0.25 0.75


The Maximax Criterion

• The maximax criterion is the decision criterion for the eternal optimist.

• It focuses only on the best that can happen.

• Procedure:– Identify the maximum payoff from any state of nature for each alternative.

– Find the maximum of these maximum payoffs and choose this alternative.

State of Nature

Alternative Oil Dry Maximum in Row

Drill for oil 700 –100 700 Maximax

Sell the land 90 90 90


The Maximin Criterion

• The maximin criterion is the decision criterion for the total pessimist.

• It focuses only on the worst that can happen.

• Procedure:– Identify the minimum payoff from any state of nature for each alternative.

– Find the maximum of these minimum payoffs and choose this alternative.

State of Nature

Alternative Oil Dry Minimum in Row

Drill for oil 700 –100 –100

Sell the land 90 90 90 Maximin


The Maximum Likelihood Criterion

• The maximum likelihood criterion focuses on the most likely state of nature.

• Procedure:– Identify the state of nature with the largest prior probability

– Choose the decision alternative that has the largest payoff for this state of nature.

State of Nature

Alternative Oil Dry

Drill for oil 700 –100 –100

Sell the land 90 90 90 Step 2: Maximum

Prior probability 0.25 0.75

Step 1: Maximum


Bayes’ Decision Rule

• Bayes’ decision rule directly uses the prior probabilities.

• Procedure:– For each decision alternative, calculate the weighted average of its payoff by

multiplying each payoff by the prior probability and summing these products. This is the expected payoff (EP).

– Choose the decision alternative that has the largest expected payoff.

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A B C D E F

Bayes' Decision Rule for the Goferbroke Co.

Payoff Table ExpectedAlternative Oil Dry Payoff

Drill 700 -100 100Sell 90 90 90

Prior Probability 0.25 0.75

State of Nature



• Features of Bayes’ Decision Rule– It accounts for all the states of nature and their probabilities.

– The expected payoff can be interpreted as what the average payoff would become if the same situation were repeated many times. Therefore, on average, repeatedly applying Bayes’ decision rule to make decisions will lead to larger payoffs in the long run than any other criterion.

• Criticisms of Bayes’ Decision Rule– There usually is considerable uncertainty involved in assigning values to the prior

probabilities.

– Prior probabilities inherently are at least largely subjective in nature, whereas sound decision making should be based on objective data and procedures.

– It ignores typical aversion to risk. By focusing on average outcomes, expected (monetary) payoffs ignore the effect that the amount of variability in the possible outcomes should have on decision making.


Decision Trees

• A decision tree can apply Bayes’ decision rule while displaying and analyzing the problem graphically.

• A decision tree consists of nodes and branches.– A decision node, represented by a square, indicates a decision to be made. The

branches represent the possible decisions.

– An event node, represented by a circle, indicates a random event. The branches represent the possible outcomes of the random event.


Decision Tree for Goferbroke

A

B

Payoff

-100

90

700Oil (0.25)

Dry (0.75)

Drill

Sell


Using TreePlan

TreePlan, an Excel add-in developed by Professor Michael Middleton, can be used to construct and analyze decision trees on a spreadsheet.

1. Choose Decision Tree under the Tools menu.

2. Click on New Tree, and it will draw a default tree with a single decision node and two branches, as shown below.

3. The labels in D2 and D7 (originally Decision 1 and Decision 2) can be replaced by more descriptive names (e.g., Drill and Sell).

123456789

A B C D E F G

Drill0

0 01

0Sell

00 0


Using TreePlan

4. To replace a node (such as the terminal node of the drill branch in F3) by a different type of node (e.g., an event node), click on the cell containing the node, choose Decision Tree again from the Tools menu, and select “Change to event node”.


Using TreePlan

5. Enter the correct probabilities in H1 and H6.

6. Enter the partial payoffs for each decision and event in D6, D14, H4, and H9.

1234567891011121314

A B C D E F G H I J K0.25

Oil700

Drill 800 700

-100 100 0.75Dry

-1001 0 -100

100

Sell90

90 90


TreePlan Results

• The numbers inside each decision node indicate which branch should be chosen (assuming the branches are numbered consecutively from top to bottom).

• The numbers to the right of each terminal node is the payoff if that node is reached.

• The number 100 in cells A10 and E6 is the expected payoff at those stages in the process.

1234567891011121314


Oil700

Drill 800 700

-100 100 0.75Dry

-1001 0 -100

100

Sell90

90 90


Consolidate the Data and Results

1234567891011121314151617181920212223242526


Oil700

Drill 800 700

-100 100 0.75Dry

-1001 0 -100

100

Sell90

90 90

DataCost of Drilling 100Revenue if Oil 800

Revenue if Sell 90Revenue if Dry 0

Probability Of Oil 0.25

Action Drill

Expected Payoff 100


Sensitivity Analysis: Prior Probability of Oil = 0.15

1234567891011121314151617181920212223242526


Oil700

Drill 800 700

-100 20 0.85Dry

-1002 0 -100

90

Sell90

90 90




Action Sell

Expected Payoff 90


Sensitivity Analysis: Prior Probability of Oil = 0.35

1234567891011121314151617181920212223242526


Oil700

Drill 800 700

-100 180 0.65Dry

-1001 0 -100

180

Sell90

90 90




Action Drill

Expected Payoff 180


Using Data Tables to Do Sensitivity Analysis

1234567891011121314151617181920212223242526272829

A B C D E F G H I J K L M0.25

Oil700

Drill 800 700

-100 100 0.75Dry

-1001 0 -100

100

Sell90

90 90

Probability ExpectedData of Oil Action Payoff

Cost of Drilling 100 Drill 100Revenue if Oil 800 0.15

Revenue if Sell 90 0.17Revenue if Dry 0 0.19

Probability Of Oil 0.25 0.210.23

Action Drill 0.250.27

Expected Payoff 100 0.290.310.330.35

Select thesecells (I18:K29), before choosing Table from the Data menu.


Data Table ResultsThe Effect of Changing the Prior Probability of Oil

1617181920212223242526272829

I J KProbability Expected

of Oil Action PayoffDrill 100

0.15 Sell 900.17 Sell 900.19 Sell 900.21 Sell 900.23 Sell 900.25 Drill 1000.27 Drill 1160.29 Drill 1320.31 Drill 1480.33 Drill 1640.35 Drill 180


Checking Whether to Obtain More Information

• Might it be worthwhile to spend money for more information to obtain better estimates?

• A quick way to check is to pretend that it is possible to actually determine the true state of nature (“perfect information”).

• EP (with perfect information) = Expected payoff if the decision could be made after learning the true state of nature.

• EP (without perfect information) = Expected payoff from applying Bayes’ decision rule with the original prior probabilities.

• The expected value of perfect information is thenEVPI = EP (with perfect information) – EP (without perfect

information).


Expected Payoff with Perfect Information

34567891011

B C DPayoff Table

Alternative Oil DryDrill 700 -100Sell 90 90

Maximum Payoff 700 90


EP (with perfect info) 242.5

State of Nature


Expected Payoff with Perfect Information

12345678910111213141516171819

A B C D E F G H I J K

Drill0.25 700

Oil 700 7001

0 700Sell

9090 90

242.5Drill

0.75 -100Dry -100 -100

20 90

Sell90

90 90


Using New Information to Update the Probabilities

• The prior probabilities of the possible states of nature often are quite subjective in nature. They may only be rough estimates.

• It is frequently possible to do additional testing or surveying (at some expense) to improve these estimates. The improved estimates are called posterior probabilities.


Seismic Survey for Goferbroke

• Goferbroke can obtain improved estimates of the chance of oil by conducting a detailed seismic survey of the land, at a cost of $30,000.

• Possible findings from a seismic survey:– FSS: Favorable seismic soundings; oil is fairly likely.

– USS: Unfavorable seismic soundings; oil is quite unlikely.

• P(finding | state) = Probability that the indicated finding will occur,given that the state of nature is the indicated one.

P(finding | state)

State of Nature Favorable (FSS) Unfavorable (USS)

Oil P(FSS | Oil) = 0.6 P(USS | Oil) = 0.4

Dry P(FSS | Dry) = 0.2 P(USS | Dry) = 0.8


Calculating Joint Probabilities

• Each combination of a state of nature and a finding will have a joint probability determined by the following formula:

P(state and finding) = P(state) P(finding | state)

• P(Oil and FSS) = P(Oil) P(FSS | Oil) = (0.25)(0.6) = 0.15.

• P(Oil and USS) = P(Oil) P(USS | Oil) = (0.25)(0.4) = 0.1.

• P(Dry and FSS) = P(Dry) P(FSS | Dry) = (0.75)(0.2) = 0.15.

• P(Dry and USS) = P(Dry) P(USS | Dry) = (0.75)(0.8) = 0.6.


Probabilities of Each Finding

• Given the joint probabilities of both a particular state of nature and a particular finding, the next step is to use these probabilities to find each probability of just a particular finding, without specifying the state of nature.

P(finding) = P(Oil and finding) + P(Dry and finding)

• P(FSS) = 0.15 + 0.15 = 0.3.

• P(USS) = 0.1 + 0.6 = 0.7.


Calculating the Posterior Probabilities

• The posterior probabilities give the probability of a particular state of nature, given a particular finding from the seismic survey.

P(state | finding) = P(state and finding) / P(finding)

• P(Oil | FSS) = 0.15 / 0.3 = 0.5.

• P(Oil | USS) = 0.1 / 0.7 = 0.14.

• P(Dry | FSS) = 0.15 / 0.3 = 0.5.

• P(Dry | USS) = 0.6 / 0.7 = 0.86.


Probability Tree Diagram

0.25(0.6) = 0.15

Oil and FSS Oil, given FSS

0.25(0.4) = 0.1Oil and USS

0.75(0.2) = 0.15

0.75(0.8) = 0.6Dry and USS

Dry and FSS

Dry, given USS

Dry, given FSS

Oil, given USS

= 0.50.15 0.3

0.1 0.7

= 0.14

0.15 0.3

= 0.5

0.6 0.7

= 0.86

PriorProbabilities

P(state)

Conditional Probabilities

P(finding | state)

Joint Probabilities

P(state and finding)

Posterior Probabilities

P(state | finding)

Unconditional probabilities: P(FSS) = 0.15 + 0.15 = 0.3

P(finding) P(USS) = 0.1 + 0.6 = 0.7

0.6 FSS, given Oil

0.4 USS, given Oil

0.2 FSS, given Dry

0.8 USS, given Dry

0.25 Oil

0.75 Dry


Template for Posterior Probabilities

345678910111213141516171819

B C D E F G HData:

State of PriorNature Probability FSS USS

Oil 0.25 0.6 0.4Dry 0.75 0.2 0.8

PosteriorProbabilities:

Finding P(Finding) Oil DryFSS 0.3 0.5 0.5USS 0.7 0.1429 0.8571

P(State | Finding)State of Nature

P(Finding | State)Finding


Decision Tree for the Full Goferbroke Co. Problem

a

b

c

d

e

f

g

h

Do seismic survey

No seismic survey

Unfavorable

Favorable

Drill

Sell

Drill

Sell

Oil

Dry

Oil

Dry

Oil

Dry

Sell

Drill


Decision Tree with Probabilities and Payoffs

a

b

c

d

e

f

g

h

Payoff

670

-130

60

670

-130

60

700

-100

90

Do seismic survey

No seismic survey

Unfavorable

Favorable

Drill-100

90Sell

Drill-100

90Sell

Drill-100

90Sell

Oil (0.143)800

0Dry(0.857)

Oil (0.5)800

0

Dry (0.5)

Oil (0.25)800

Dry (0.75)

0

(0.3)-30

0

0

0


The Final Decision Tree

a

b

c

d

e

f

g

h

Payoff

670

-130

60

670

-130

60

700

-100

90

100

270

60

123

123

-15.7

270

100

Do seismic survey

No seismic survey

-30

0

Unfavorable

0

0

Favorable (0.3)

Drill-100

90Sell

Drill -100

90Sell

Drill-100

90Sell

Oil (0.143)800

0Dry (0.857)

Oil (0.5)800

0Dry (0.5)

Oil (0.25)

800

0

Dry (0.75)


TreePlan for the Full Goferbroke Co. Problem12345678910111213141516171819202122232425262728293031323334353637383940414243444546

A B C D E F G H I J K L M N O P Q R S

Decision Tree for Goferbroke Co. Problem (With Survey)

0.143Oil

670Drill 800 670

-100 -15.714 0.8570.7 Dry

Unfavorable -1302 0 -130

0 60

Sell60

90 60Do Survey

0.5-30 123 Oil

670Drill 800 670

-100 270 0.50.3 Dry

Favorable -1301 0 -130

0 270

1 Sell123 60

90 60

0.25Oil

700Drill 800 700

-100 100 0.75Dry

No Survey -1001 0 -100

0 100

Sell90

90 90


Organizing the Spreadsheet for Sensitivity Analysis345678910111213141516171819202122232425262728293031323334353637383940414243444546

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z AA0.143 Data

Oil Cost of Survey 30670 Cost of Drilling 100

Drill 800 670 Revenue if Oil 800Revenue if Sell 90

-100 -15.714 0.857 Revenue if Dry 00.7 Dry Prior Probability Of Oil 0.25

Unfavorable -130 P(FSS|Oil) 0.62 0 -130 P(USS|Dry) 0.8

0 60

Sell Action60 Do Survey? Yes

90 60Do Survey If No If Yes

0.5-30 123 Oil Drill Drill If Favorable

670 Sell If UnfavorableDrill 800 670

-100 270 0.50.3 Dry Expected Payoff

Favorable -130 ($thousands)1 0 -130 123

0 270

1 Sell123 60 Data:

90 60 State of PriorNature Probability FSS USS

0.25 Oil 0.25 0.6 0.4Oil Dry 0.75 0.2 0.8

700Drill 800 700

-100 100 0.75Dry Posterior

No Survey -100 Probabilities:1 0 -100 Finding P(Finding) Oil Dry

0 100 FSS 0.3 0.5 0.5USS 0.7 0.143 0.857

Sell90

90 90

P(State | Finding)

Finding

State of Nature

P(Finding | State)


The Plot Option of SensIt

34567891011121314151617181920212223242526

U V W X YData

Cost of Survey 30Cost of Drilling 100Revenue if Oil 800


Prior Probability Of Oil 0.25P(FSS|Oil) 0.6

P(USS|Dry) 0.8

ActionDo Survey? Yes

If No If Yes

Drill Drill If FavorableSell If Unfavorable

Expected Payoff($thousands)

123


SensIt Plot

Sensit - Sensitivity Analysis - Plot

0

100

200

300

400

500

600

700

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Prior Probability Of Oil

Exp

ecte

d P

ayof

f


Optimal Policy

Let p = Prior probability of oil

If p ≤ 0.168, then sell the land (no seismic survey).

If 0.169 ≤ p ≤ 0.308, then do the survey; drill if favorable, sell if not.

If p ≥ 0.309, then drill for oil (no seismic survey).


The Spider Option of SensIt

34567891011121314151617181920212223242526

U V W X YData

Cost of Survey 30Cost of Drilling 100Revenue if Oil 800



P(USS|Dry) 0.8


If No If Yes



123


SensIt Spider Graph

Sensit - Sensitivity Analysis - Spider

110

112

114

116

118

120

122

124

126

128

130

132

134

136

90% 92% 94% 96% 98% 100% 102% 104% 106% 108% 110%

% Change in Input Value

Exp

ect

ed

Pa

yoff

Va

lue

Cost of SurveyCost of DrillingRevenue if OilRevenue if Sell


The Tornado Option of SensIt

34567891011121314151617181920212223242526

U V W X YData Low Base High

Cost of Survey 30 28 30 32Cost of Drilling 100 75 100 140Revenue if Oil 800 600 800 1000

Revenue if Sell 90 85 90 95Revenue if Dry 0


P(USS|Dry) 0.8


If No If Yes



123


SensIt Tornado Diagram

Sensit - Sensitivity Analysis - Tornado

600

75

85

28

1000

140

95

32

90 100 110 120 130 140 150 160

Revenue if Oil

Cost of Drilling

Revenue if Sell

Cost of Survey

Expected Payoff


Using Utilities to Better Reflect the Values of Payoffs

• Thus far, when applying Bayes’ decision rule, we have assumed that the expected payoff in monetary terms is the appropriate measure.

• In many situations, this is inappropriate.

• Suppose an individual is offered the following choice:– Accept a 50-50 chance of winning $100,000.

– Receive $40,000 with certainty.

• Many would pick $40,000, even though the expected payoff on the 50-50 chance of winning $100,000 is $50,000. This is because of risk aversion.

• A utility function for money is a way of transforming monetary values to an appropriate scale that reflects a decision maker’s preferences (e.g., aversion to risk).


A Typical Utility Function for Money

0

0.25

0.5

0.75

1

$10,000 $30,000 $60,000 $100,000 M

U(M)


Shape of Utility Functions

U(M)

M M M

U(M) U(M)

(a) Risk averse (b) Risk seeker (c) Risk neutral


Utility Functions

• When a utility function for money is incorporated into a decision analysis approach, it must be constructed to fit the current preferences and values of the decision maker.

• Fundamental Property: Under the assumptions of utility theory, the decision maker’s utility function for money has the property that the decision maker is indifferent between two alternatives if the two alternatives have the same expected utility.

• When the decision maker’s utility function for money is used, Bayes’ decision rule replaces monetary payoffs by the corresponding utilities.

• The optimal decision (or series of decisions) is the one that maximizes the expected utility.


Illustration of Fundamental Property

0

0.25

0.5

0.75

1

$10,000 $30,000 $60,000 $100,000 M

U(M)

By the fundamental property, a decision maker with the utility function below-right will be indifferent between each of the three pairs of alternatives below-left.

• 25% chance of $100,000• $10,000 for sureBoth have E(Utility) = 0.25.




The Lottery Procedure

1. We are given three possible monetary payoffs—M1, M2, M3 (M1 < M2 < M3). The utility is known for two of them, and we wish to find the utility for the third.

2. The decision maker is offered the following two alternatives:a) Obtain a payoff of M3 with probability p.

Obtain a payoff of M1 with probability (1–p).

b) Definitely obtain a payoff of M2.

3. What value of p makes you indifferent between the two alternatives?

4. Using this value of p, write the fundamental property equation,E(utility for a) = E(utility for b)

sop U(M3) + (1–p) U(M1) = U(M2).

5. Solve this equation for the unknown utility.


Procedure for Constructing a Utility Function

1. List all the possible monetary payoffs for the problem, including 0.

2. Set U(0) = 0 and then arbitrarily choose a utility value for one other payoff.

3. Choose three of the payoffs where the utility is known for two of them.

4. Apply the lottery procedure to find the utility for the third payoff.

5. Repeat steps 3 and 4 for as many other payoffs with unknown utilities as desired.

6. Plot the utilities found on a graph of the utility U(M) versus the payoff M. Draw a smooth curve through these points to obtain the utility function.


Generating the Utility Function for Max Flyer

• The possible monetary payoffs in the Goferbroke Co. problem are –130, –100, 0, 60, 90, 670, and 700 (all in $thousands).

• Set U(0) = 0.

• Arbitrarily set U(–130) = –150.


Finding U(700)

• The known utilities are U(–130) = –150 and U(0) = 0.The unknown utility is U(700).

• Consider the following two alternatives:a) Obtain a payoff of 700 with probability p.

Obtain a payoff of –130 with probability (1–p).

b) Definitely obtain a payoff of 0.

• What value of p makes you indifferent between these two alternatives?Max chooses p = 0.2.

• By the fundamental property of utility functions, the expected utilities of the two alternatives must be equal, so

pU(700) + (1–p)U(–130) = U(0)0.2U(700) + 0.8(–150) = 00.2U(700) – 120 = 00.2U(700) = 120U(700) = 600


Finding U(–100)

• The known utilities are U(–130) = –150 and U(0) = 0.The unknown utility is U(–100).


Obtain a payoff of –130 with probability (1–p).

b) Definitely obtain a payoff of –100.



pU(0) + (1–p)U(–130) = U(–100)0.3(0) + 0.7(–150) = U(–100)U(–100) = –105


Finding U(90)

• The known utilities are U(700) = 600 and U(0) = 0.The unknown utility is U(90).


Obtain a payoff of 0 with probability (1–p).

b) Definitely obtain a payoff of 90.



pU(700) + (1–p)U(0) = U(90)0.15(600) + 0.85(0) = U(90)U(90) = 90


Max’s Utility Function for Money

0

100

200

300

400

500

600

700

100 200 300 400 500 600 700-100-200

-100

-200

M

Thousands of dollars

U(M)

monetary value line

utility function


Utilities for the Goferbroke Co. Problem

Monetary Payoff, M Utility, U(M)

–130 –150

–100 –105

0 0

60 60

90 90

670 580

700 600


Decision Tree with Utilities345678910111213141516171819202122232425262728293031323334353637383940414243444546

A B C D E F G H I J K L M N O P Q R S0.143

Oil580

Drill 580 580

0 -45.61 0.8570.7 Dry

Unfavorable -1502 -150 -150

0 60

Sell60

60 60Do Survey

0.50 106.5 Oil

580Drill 580 580

0 215 0.50.3 Dry

Favorable -1501 -150 -150

0 215

1 Sell106.5 60

60 60

0.25Oil

600Drill 600 600

0 71.25 0.75Dry

No Survey -1052 -105 -105

0 90

Sell90

90 90


Exponential Utility Function

• The procedure for constructing U(M) requires making many difficult decisions about probabilities.

• An alternative approach assumes a certain form for the utility function and adjusts this form to fit the decision maker as closely as possible.

• A popular form is the exponential utility function

U(M) = R (1 – e–M/R)

where R is the decision maker’s risk tolerance.

• An easy way to estimate R is to pick the value that makes you indifferent between the following two alternatives:

a) A 50-50 gamble where you gain R dollars with probability 0.5 and lose R/2 dollars with probability 0.5.

b) Neither gain nor lose anything.


Using TreePlan with an Exponential Utility Function

• Specify the value of R in a cell on the spreadsheet.

• Give the cell a range name of RT (TreePlan refers to this term as the risk tolerance).

• Click on the Option button in the TreePlan dialogue box and select the “Use Exponential Utility Function” option.


Decision Tree with an Exponential Utility Function3456789101112131415161718192021222324252627282930313233343536373839404142434445464748


Oil670

Drill 800 6700.602

-100 -57.052 0.8570.7 -0.0815 Dry

Unfavorable -1302 0 -130

0 60 -0.1960.0791

Sell60

90 60Do Survey 0.0791

0.5-30 90.0036 Oil

0.1163 670Drill 800 670

0.602-100 165.23116 0.5

0.3 0.203 DryFavorable -130

1 0 -1300 165.231 -0.196

0.2031 Sell

90 600.1163 90 60

0.07910.25

Oil700

Drill 800 7000.618

-100 32.7511 0.750.0440 Dry

No Survey -1002 0 -100

0 90 -0.1470.11629

Sell90

90 900.1163

Risk Tolerance (RT) 728


Decisions Under Certainty

• State of nature is certain (one state).

• Select decision that yields highest return (e.g., linear programming, integer programming).

• Examples:– Product mix

– Diet problem

– Distribution

– Scheduling


Decisions Under Uncertainty (or Risk)

• State of nature is uncertain (several possible states)

• Examples– Drilling for oil

• Uncertainty: Oil found? How much? How deep? Selling Price?

• Decision: Drill or not?

– Developing a new product

• Uncertainty: R&D Cost, demand, etc.

• Decisions: Design, quantity, produce or not?

– Newsvendor problem

• Uncertainty: Demand

• Decision: Stocking levels

– Producing a movie

• Uncertainty: Cost, gross, etc.

• Decisions: Develop? Arnold or Keanu?


Oil Drilling Problem

• Consider the problem faced by an oil company that is trying to decide whether to drill an exploratory oil well on a given site.

• Drilling costs $200,000.

• If oil is found, it is worth $800,000.

• If the well is dry, it is worth nothing.

State of Nature

Decision Wet Dry

Drill 600 –200

Do not drill 0 0


Decision Criteria

Which decision is best?

• “Optimist”

• “Pessimist”

• “Second–Guesser”

• “Joe Average”

State of Nature

Decision Wet Dry

Drill 600 –200

Do not drill 0 0



• Suppose that the oil company estimates that the probability that the site is “Wet” is 40%.

State of Nature

Decision Wet Dry

Drill 600 –200

Do not drill 0 0


• Expected value of payoff (Drill) = (0.4)(600) + (0.6)(–200) = 120

• Expected value of payoff (Do not drill) = (0.4)(0) + (0.6)(0) = 0

Bayes’ Decision Rule: Choose the decision that maximizes the expected payoff (Drill).


Features of Bayes’ Decision Rule

• Accounts not only for the set of outcomes, but also their probabilities.

• Represents the average monetary outcome if the situation were repeated indefinitely.

• Can handle complicated situations involving multiple related risks.


Using a Decision Tree to Analyze Oil Drilling Problem

600

-200

0

Wet

Dry

Drill

Do not drill

0.6

0.4

Folding Back:

• At each event node (circle): calculate expected value (SUMPRODUCT of payoffs and probabilities for each branch).

• At each decision node (square): choose “best” branch (maximum value).


Using TreePlan to Analyze Oil Drilling Problem

1. Choose Decision Tree under the Tools menu.

2. Click on “New Tree” and it will draw a default tree with a single decision node and two branches, as shown below.

3. Label each branch. Replace “Decision 1” with “Drill” (cell D2). Replace “Decision 2” with “Do not drill” (cell D7).

4. To replace the terminal node of the drill branch with an event node, click on the terminal node (cell F3) and then choose Decision Tree under the Tools menu. Click on “Change to event node,” choose two branches, then click OK.

123456789

A B C D E F G

Decision 10

0 01

0Decision 2

00 0


Using TreePlan to Analyze Oil Drilling Problem

5. Change the labels “Event 3” and “Event 4” to “Wet” and “Dry”, respectively.

6. Change the default probabilities (cells H1 and H6) from 0.5 and 0.5 to the correct values of 0.4 and 0.6.

7. Enter the partial payoffs under each branch: (-200) for “Drill” (D6), 0 for “Do not drill” (D14), 800 for “Wet” (H4), and 0 for “Dry” (H9). The terminal value cash flows are calculated automatically from the partial cash flows.

1234567891011121314


Event 30

Drill 0 0

0 0 0.5Event 4

01 0 0

0

Do not drill0

0 0


Final Decision Tree

1234567891011121314


Wet600

Drill 800 600

-200 120 0.6Dry

-2001 0 -200

120

Do not drill0

0 0


Features of TreePlan

• Terminal values (payoff) are calculated automatically from the partial payoffs (K3 = D6+H4, K8 = D6+H9, K13 = D14).

• Foldback values are calculated automatically (I4 = K3, I9 = K8, E6 = H1*I4 + H6*I9, E14 = K13, A10 = Max(E6,E14)).

• Optimal decisions are indicated inside decision node squares (labeled by branch number from top to bottom, e.g., branch #1 = Drill, branch #2 = Do not drill).

• Changes in the tree can be made by clicking on a node and choosing Decision Tree under the Tools menu (change type of node, # of branches, etc.)

• Clicking “Options…” in the Decision Tree dialogue box allows the choice of Maximize Profit or Minimize Cost.


Making Sequential Decisions

• Consider a pharmaceutical company that is considering developing an anticlotting drug.

• They are considering two approaches– A biochemical approach (more likely to be successful)– A biogenetic approach (more radical)

• While the biogenetic approach is not nearly as likely to succeed, if would likely capture a much larger portion of the market if it did.

R&D Choice Investment OutcomesProfit

(excluding R&D) Probability

Biochemical $10 million Large successSmall success

$90 million$50 million

0.70.3

Biogenetic $20 million SuccessFailure

$200 million$0 million

0.20.8


Biochemical vs. Biogenetic

12345678910111213141516171819


Large Success80

Biochemical 90 80

-10 68 0.3Small Success

4050 40

168 0.2

Success180

Biogenetic 200 180

-20 20 0.8Failure

-200 -20


Simultaneous Development

1234567891011121314151617181920212223242526272829

A B C D E F G H I J K L M N O

Market BC0.14 60

Large Success (BC), Success (BG) 90 602

0 170Market BG

170200 170

Market BC0.06 20

Small Success (BC), Success (BG) 50 20Simultaneous Development 2

1 0 17072.4 -30 72.4 Market BG

170200 170

0.56Large Success (BC), Failure (BG) Market BC

1 600 60 90 60

0.24Small Success (BC), Failure (BG) Market BC

1 200 20 50 20


Biochemical First123456789101112131415161718192021222324252627282930313233343536373839

A B C D E F G H I J K L M N O P Q R S T U V W

Market BC80

90 80

0.7Large Success (BC) Market BC

2 0.2 600 82 Success (BG) 90 60

20 170

Market BGPursue BG 170

200 170-20 82

0.8Biochemical First Failure (BG) Market BC

1 1 6072.4 -10 72.4 0 60 90 60

Market BC40

50 40

0.3Small Success (BC) Market BC

2 0.2 200 50 Success (BG) 50 20

20 170

Market BGPursue Biogenetic 170

200 170-20 50

0.8Failure (BG) Market BC

1 200 20 50 20


Biogenetic First123456789101112131415161718192021222324252627282930313233343536373839

A B C D E F G H I J K L M N O P Q R S T U V W

Market BC0.7 60

Large Success (BC) 90 602

0 170Market BG

170Pursue BC 200 170

-10 170Market BC

0.3 200.2 Small Success (BC) 50 20

Success (BG) 22 0 170

0 180 Market BG170

200 170

Market BG180

Biogenetic First 200 1801

74.4 -20 74.4 0.7Large Success (BC) Market BC

1 60Pursue BC 0 60 90 60

-10 48 0.30.8 Small Success (BC) Market BC

Failure (BG) 1 201 0 20 50 20

0 48

Don't Pursue BC-20

0 -20


Incorporating New Information

• Often, a preliminary study can be done to better determine the true state of nature.

• Examples:– Market surveys

– Test marketing

– Seismic testing (for oil)

Question: What is the value of this information?


Oil Drilling Problem

Consider again the problem faced by an oil company that is trying to decide whether to drill an exploratory oil well on a given site. Drilling costs $200,000. If oil is found, it is worth $800,000. If the well is dry, it is worth nothing. The prior probability that the site is wet is estimated at 40%.

State of Nature

Decision Wet Dry

Drill 600 –200

Do not drill 0 0


• Expected Payoff (Drill) = (0.4)(600) + (0.6)(–200) = 120

• Expected Payoff (Do not drill) = (0.4)(0) + (0.6)(0) = 0


Expected Value of Perfect Information (EVPI)

State of Nature

Decision Wet Dry

Drill 600 –200

Do not drill 0 0


Suppose they had a test that could predict ahead of time whether the side would be wet or dry.

• Expected Payoff = (0.4)(600) + (0.6)(0) = 240

• Expected Value of Perfect Information (EVPI)= Expected Payoff (with perfect info) – Expected Payoff (without

info)= 240 – 120= 120


Using TreePlan to Calculate EVPI

2345678910111213141516171819

A B C D E F G H I J KDrill

0.4 600Wet 600 600

10 600

Do not drill0

0 0

240Drill

0.6 -200Dry -200 -200

20 0

Do not drill0

0 0


Imperfect Information (Seismic Test)

Suppose a seismic test is available that would better (but not perfectly) indicate whether or not the site was wet or dry.

– Good result usually means the site is wet (but not always)

– Bad results usually means the site is dry (but not always)

Record of 100 Past Seismic Test Sites

SeismicResult

Actual State of Nature

Wet (W) Dry (D) Total

Good (G) 30 20 50

Bad (B) 10 40 50

Total 40 60 100


Decision Tree with Seismic Test

123456789

10111213141516171819202122232425262728

F G H I J K L M N O P Q R SP(W | G) = ?Wet

600Drill

P(D | G) = ?P(G) = ? DryGood Test (G) -200

Do not drill0

P(W | B) = ?Wet

600Drill

P(D | B) = ?P(B) = ? DryBad Test (B) -200

Do not drill0


Conditional Probabilities

• Need probabilities of each test result:– P(G) = 50 / 100 = 0.5

– P(B) = 50 / 100 = 0.5

• Need conditional probabilities of each state of nature, given a test result:– P(W | G) = 30 / 50 = 0.6

– P(D | G) = 20 / 50 = 0.4

– P(W | B) = 10 / 50 = 0.2

– P(D | B) = 40 / 50 = 0.8

SeismicResult

Actual State of Nature

Wet (W) Dry (D) Total

Good (G) 30 20 50

Bad (B) 10 40 50

Total 40 60 100


Expected Value of Sample Information (EVSI)123456789

1011121314151617181920212223242526272829303132333435363738394041424344


Wet600

Drill 800 600

-200 280 0.40.5 Dry

Good Test (G) -2001 0 -200

0 280

Do not drill0

0 0Do Seismic Test

0.20 140 Wet

600Drill 800 600

-200 -40 0.80.5 Dry

Bad Test (B) -2002 0 -200

0 0

1 Do not drill140 0

0 0

0.4Wet

600Drill 800 600

-200 120 0.6Dry

Forego test -2001 0 -200

0 120

Do not drill0

0 0

Expected Value of Sample Information

= EVSI

= 140 – 120

= 20.


Revising Probabilities

• Suppose they don’t have the “Record of Past 100 Seismic Test Sites”.

• Vendor of test certifies:– Wet sites test “good” three quarters of the time.

– Dry sites test “bad” two thirds of the time

P(G | W) = 3/4 P(B | W) = 1/4

P(B | D) = 2/3 P(G | D) = 1/3

Is this the information needed in the decision tree?


Revising Probabilities (Probability Tree Diagram)Prior

ProbabilitiesConditionalProbabilities

JointProbabilities

PosteriorProbabilities

Wet0.4

Dry0.6

Good, given Wet

Bad, given Wet

0.75

0.25

Good, given Dry

Bad, given Dry

0.333

0.667

Good and Wet

Bad and Wet

Good and Dry

Bad and Dry

Wet, given Good

Wet, given Bad

Dry, given Good

Dry, given Bad

(0.4)(0.75) = 0.3

(0.4)(0.25) = 0.1

(0.6)(0.33) = 0.2

(0.6)(0.67) = 0.2

0.3 / 0.5 = 0.6

0.1 / 0.5 = 0.2

0.2 / 0.5 = 0.4

0.4 / 0.5 = 0.8

P(Good) = 0.3 + 0.2 = 0.5

P(Bad) = 0.1 + 0.4 = 0.5

P(state) P(finding | state) P(finding & state) P(State | Finding)


Template for Posterior Probabilities

Template available on textbook CD.

345678910111213141516171819

B C D E F G HData:

State of PriorNature Probability Good Bad

Wet 0.4 0.75 0.25Dry 0.6 0.333 0.667

PosteriorProbabilities:

Finding P(Finding) Wet DryGood 0.5 0.6 0.4Bad 0.5 0.2 0.8

P(State | Finding)State of Nature

P(Finding | State)Finding


Risk Attitude

• Consider the following coin-toss gambles. How much would you sell each of these gambles for?

• Heads: You win $200Tails: You lose $0

• Heads: You win $300Tails: You lose $100

• Heads: You win $20,000Tails: You lose $0

• Heads: You win $30,000Tails: You lose $10,000


Demand for Insurance

• House Value = $150,000

• Insurance Premium = $500

• Probability of fire destroying house (in one year) = 1 / 1,000

Question: Should you buy insurance?

234567891011121314

A B C D E F G H I J KBuy Insurance

-500-500 -500

2 0.001-150 Fire

-150000Self-Insure -150000 -150000

0 -150 0.999No Fire

00 0


Utilities and Risk Aversion

-120 0 200 600-200

0.25

0.50

0.75

1.00

0

Monetary Values (Thousands of Dollars)

Utility

Utility Curve

Payoff Utility

$600,000 1.0

200,000 0.75

0 0.50

–120,000 0.25

–200,000 0


Oil Drilling Problem (Risk Aversion)

Risk Neutral: Risk Averse:

$600

- $200

Wet

Dry

0.4Drill

Do not drill

0.6

$0

120

120

1

U($600)

U(-$200)

Wet

Dry

0.4Drill

Do not drill

0.6

U($0)

= 1

= 0

= 0.5

0.4

0.5

2


Creating a Utility Function(Equivalent Lottery Method)

1. Set U(Min) = 0.

2. Set U(Max) = 1.

3. To find U(x):Choose p such that you are indifferent between the following:

a) A payment of x for sure.

b) A payment of Max with probability p and a payment of Min with probability 1–p.

4. U(x) = p.


Equivalent Lottery Method

• Uncertain situation: –$200 in worst case U(–$200) = 0$1,800 in best case U($1,800) = 1

• U($800) =

• U($200) =

• U($400) =

• U($600) =

$1800

-$200

p

1-p

$x

Gamble

Certain Equivalent

EU = p

U($1800) = 1

U(-$200) = 0

U = ?


Utility CurveUtility1.0

0.6

0.4

0.2

-$200

Monetary Value

$600 $1400$200 $10000

0.8

$1800


Biochemical vs. Biogenetic First (Expected Payoff)

1234567891011121314151617181920212223242526272829


Success180

200 180

0.7Biogenetic First Large Success

60-20 74.4 Pursue Biochemical 90 60

-10 48 0.30.8 Small Success

Failure 201 50 20

0 481

74.4 Don't Pursue Biochemical-20

0 -20

0.7Large Success

80Biochemical 90 80

-10 68 0.3Small Success

4050 40


Biochemical vs. Biogenetic First (with Utilities)

123456789

1011121314151617181920212223242526272829


Success1

200 1


0.7-20 0.688 Pursue Biochemical 90 0.7

-10 0.61 0.30.8 Small Success

Failure 0.41 50 0.4

0 0.612

0.74 Don't Pursue Biochemical0

0 0

0.7Large Success

0.8Biochemical 90 0.8

-10 0.74 0.3Small Success

0.650 0.6



Choose R so that you are indifferent between the following:

U(M) = R(1 – e–M / R)

$R

-$R/2

0.5

0.5

$0

Gamble

Certain Equivalent



U(M) = R(1 – e–M / R)

Utility

Monetary Value0


Using an Exponential Utility Function with TreePlan

• To use an exponential utility function in TreePlan, enter the R value in a cell on the spreadsheet

• Give this cell the range name RT (TreePlan calls this value the risk tolerance).

• Choose “Use Exponential Utility Function” in the dialogue box shown below (available by clicking on “Options…” in the Decision Tree dialogue box).


Biochemical vs. Biogenetic First(with Exponential Utility)

123456789

1011121314151617181920212223242526272829303132


Success180

200 1800.8347


60-20 62.1963 Pursue Biochemical 90 60

0.46311 0.45119-10 46.237 0.3

0.8 0.3702 Small SuccessFailure 20

1 50 200 46.2373 0.18127

2 0.3702166.2373 Don't Pursue Biochemical0.48437 -20

0 -20-0.2214

0.7Large Success

80Biochemical 90 80

0.55067-10 66.2373 0.3

0.48437 Small Success40

50 400.32968

RT = 100

mcgraw-hill/irwin © the mcgraw-hill companies, inc., 2003 12.1 table of contents chapter 12...

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