decision maker.ppt

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Decision Models

1

Chapter 6



6.1 Introduction to Decision

Analysis• The field of decision analysis provides a

framework for making important decisions.• Decision analysis allows us to select a decision

from a set of possible decision alternatives whenuncertainties regarding the future exist.

• The goal is to optimize the resulting payoff interms of a decision criterion.

2



6.1 Introduction to Decision

Analysis• Maximizing expected profit is a

common criterion when probabilities

can be assessed.

3

• Maximizing the decision maker’s utility

function is the mechanism used when riskis factored into the decision making

process.



6.2 Payoff Table Analysis

• Payoff Tables

• Payoff table analysis can be applied when:• There is a finite set of discrete decision alternatives.• The outcome of a decision is a function of a single future event.

• In a Payoff table -• The rows correspond to the possible decision alternatives.• The columns correspond to the possible future events.• Events ( states of nature ) are mutually exclusive and collectively

exhaustive.• The table entries are the payoffs. 4



TOM BROWN INVESTMENT

DECISION• Tom Brown has inherited $1000.• He has to decide how to invest the money for

one year.• A broker has suggested five potential

investments.• Gold• Junk Bond• Growth Stock• Certificate of Deposit• Stock Option Hedge

5



TOM BROWN• The return on each investment depends on the (uncertain)

market behavior during the year.• Tom would build a payoff table to help make the investment

decision

6



TOM BROWN - Solution

• Select a decision making criterion, and apply it to the

payoff table.

7

S1 S2 S3 S4D1 p11 p12 p13 p14 D2 p21 p22 p23 P24

D3 p31 p32 p33 p34

S1 S2 S3 S4D1 p11 p12 p13 p14 D2 p21 p22 p23 P24

D3 p31 p32 p33 p34

CriterionP1P2

P3

• Construct a payoff table.

• Identify the optimal decision.

• Evaluate the solution.



The Payoff Table

8

Decision States of Nature Alternatives Large Rise Small Rise No Change Small Fall Large Fall

Gold -100 100 200 300 0

Bond 250 200 150 -100 -150

Stock 500 250 100 -200 -600

C/D account 60 60 60 60 60

Stock option 200 150 150 -200 -150

The states of nature are mutuallyexclusive and collectively exhaustive.



The Payoff Table

9

Decision States of Nature

Alternatives Large Rise Small Rise No Change Small Fall Large FallGold -100 100 200 300 0

Bond 250 200 150 -100 -150

Stock 500 250 100 -200 -600

C/D account 60 60 60 60 60

Stock option 200 150 150 -200 -150

Determine theset of possible

decisionalternatives.



The Payoff Table

10

Decision States of Nature

Alternatives Large Rise Small Rise No Change Small Fall Large FallGold -100 100 200 300 0Bond 250 200 150 -100 -150Stock 500 250 100 -200 -600

C/D account 60 60 60 60 60Stock option 200 150 150 -200 -150

The stock option alternative is dominated by the

bond alternative

250 200 150 -100 -150

-150



6.3 Decision Making Criteria

• Classifying decision-making criteria

• Decision making under certainty.• The future state-of-nature is assumed known.

• Decision making under risk.• There is some knowledge of the probability of the states of nature occurring.

• Decision making under uncertainty.• There is no knowledge about the probability of the states of nature occurring.

11



Decision Making UnderUncertainty

• The decision criteria are based on the decision maker’s attitude toward life.

• The criteria include the• Maximin Criterion - pessimistic or conservative approach.• Minimax Regret Criterion - pessimistic or conservative approach.• Maximax Criterion - optimistic or aggressive approach.• Principle of Insufficient Reasoning – no information about the

likelihood of the various states of nature.12



ec s on a ng n erUncertainty - The Maximin

Criterion• This criterion is based on the worst-case scenario.

• It fits both a pessimistic and a conservative decisionmaker’s styles.

• A pessimistic decision maker believes that the worst possible result will

always occur.• A conservative decision maker wishes to ensure a guaranteed minimum

possible payoff.

13



TOM BROWN - The Maximin Criterion

• To find an optimal decision• Record the minimum payoff across all states of nature for

each decision.

• Identify the decision with the maximum “minimum payoff.”

14

The Maximin Criterion Minimum Decisions Large Rise Small rise No Change Small Fall Large Fall Payoff Gold -100 100 200 300 0 -100Bond 250 200 150 -100 -150 -150Stock 500 250 100 -200 -600 -600C/D account 60 60 60 60 60 60

The Maximin Criterion Minimum Decisions Large Rise Small rise No Change Small Fall Large Fall Payoff

Gold -100 100 200 300 0 -100Bond 250 200 150 -100 -150 -150Stock 500 250 100 -200 -600 -600C/D account 60 60 60 60 60 60



The Maximin Criterion - spreadsheet

15

=MAX(H4:H7)

* FALSE is the range lookup argument inthe VLOOKUP function in cell B11 since thevalues in column H are not in ascendingorder

=VLOOKUP(MAX(H4:H7),H4:I7,2,FALSE)

=MIN(B4:F4)Drag to H7



The Maximin Criterion - spreadsheet

16

To enable the spreadsheet to correctly identify the optimalmaximin decision in cell B11, the labels for cells A4 through A7 are copied into cells I4 through I7 (note that column I inthe spreadsheet is hidden).

I4

Cell I4 (hidden)=A4

Drag to I7



ec s on a ng n erUncertainty - The Minimax

Regret Criterion• The Minimax Regret Criterion

•

This criterion fits both a pessimistic and aconservative decision maker approach.

• The payoff table is based on “lost opportunity,”or “regret.”

• The decision maker incurs regret by failing tochoose the “best” decision.

17



ec s on a ng n erUncertainty - The Minimax

Regret Criterion• The Minimax Regret Criterion

• To find an optimal decision, for each state of nature:• Determine the best payoff over all decisions.• Calculate the regret for each decision alternative as the difference

between its payoff value and this best payoff value.• For each decision find the maximum regret over all states of nature.• Select the decision alternative that has the minimum of these

“maximum regrets.”

18



TOM BROWN – Regret Table

19

The Payoff TableDecision Large rise Small rise No change Small fall Large fallGold -100 100 200 300 0Bond 250 200 150 -100 -150Stock 500 250 100 -200 -600C/D 60 60 60 60 60


Let us build the Regret Table

The Regret TableDecision Large rise Small rise No change Small fall Large fallGold 600 150 0 0 60Bond 250 50 50 400 210Stock 0 0 100 500 660C/D 440 190 140 240 0

Investing in Stock generates noregret when the market exhibits

a large rise



TOM BROWN – Regret Table

20



The Regret Table MaximumDecision Large rise Small rise No change Small fall Large fall RegretGold 600 150 0 0 60 600Bond 250 50 50 400 210 400Stock 0 0 100 500 660 660C/D 440 190 140 240 0 440

The Regret Table MaximumDecision Large rise Small rise No change Small fall Large fall RegretGold 600 150 0 0 60 600Bond 250 50 50 400 210 400Stock 0 0 100 500 660 660C/D 440 190 140 240 0 440

Investing in gold generates a regretof 600 when the market exhibits

a large rise

500 – (-100) = 600



The Minimax Regret -spreadsheet

21

=MAX(B$4:B$7)-B4Drag to F16

=VLOOKUP(MIN(H13:H16),H13:I16,2,FALSE)

=MIN(H13:H16)

=MAX(B14:F14)Drag to H18

Cell I13 (hidden)=A13Drag to I16



ec s on a ng n erUncertainty - The Maximax

Criterion• This criterion is based on the best possible scenario.

It fits both an optimistic and an aggressive decisionmaker.

• An optimistic decision maker believes that the bestpossible outcome will always take place regardless ofthe decision made.

• An aggressive decision maker looks for the decision withthe highest payoff (when payoff is profit).

22



• To find an optimal decision.• Find the maximum payoff for each decision alternative.• Select the decision alternative that has the maximum of the

“maximum” payoff.

23

Decision Making Under Uncertainty -

The Maximax Criterion



TOM BROWN - The Maximax Criterion

24

The Maximax Criterion MaximumDecision Large rise Small rise No change Small fall Large fall Payoff Gold -100 100 200 300 0 300

Bond 250 200 150 -100 -150 200

Stock 500 250 100 -200 -600 500

C/D 60 60 60 60 60 60



• This criterion might appeal to a decision maker who is neitherpessimistic nor optimistic.

•

It assumes all the states of nature are equally likely to occur.• The procedure to find an optimal decision.

• For each decision add all the payoffs.• Select the decision with the largest sum (for profits).

25

Decision Making Under Uncertainty -

The Principle of Insufficient Reason



TOM BROWN - Insufficient Reason

• Sum of Payoffs• Gold 600 Dollars• Bond 350 Dollars• Stock 50 Dollars• C/D 300 Dollars

• Based on this criterion the optimal decision alternative is toinvest in gold.

26



27

Decision Making Under Uncertainty –

Spreadsheet templatePayoff Table

Large Rise Small Rise No Change Small Fall Large FallGold -100 100 200 300 0Bond 250 200 150 -100 -150

Stock 500 250 100 -200 -600C/D Account 60 60 60 60 60d5d6d7d8Probability 0.2 0.3 0.3 0.1 0.1

Criteria Decision Payoff Maximin C/D Account 60Minimax Regret Bond 400Maximax Stock 500Insufficient Reason Gold 100EV Bond 130

EVPI 141

RESULTS



Decision Making Under Risk

28

• The probability estimate for the occurrence ofeach state of nature (if available) can beincorporated in the search for the optimaldecision.

• For each decision calculate its expected payoff.



Decision Making Under Risk – The Expected Value Criterion

• For each decision calculate the expected payoff as follows:

(The summation is calculated across all the states ofnature)

• Select the decision with the best expected payoff

29

Expected Payoff = (Probability)(Payoff)



TOM BROWN - The Expected Value

Criterion

30

The Expected Value Criterion ExpectedDecision Large rise Small rise No change Small fall Large fall Value

Gold -100 100 200 300 0 100Bond 250 200 150 -100 -150 130Stock 500 250 100 -200 -600 125C/D 60 60 60 60 60 60Prior Prob. 0.2 0.3 0.3 0.1 0.1

EV = (0.2)(250) + (0.3)(200) + (0.3)(150) + (0.1)(-100) + (0.1)(-150) =



When to use the expected value

approach• The expected value criterion is useful generally in two cases:

• Long run planning is appropriate, and decision situations repeat

themselves.• The decision maker is risk neutral.

31



6.4 Expected Value of Perfect

Information• The gain in expected return obtained from knowing with certainty

the future state of nature is called:

Expected Value of Perfect Information

(EVPI)

33



TOM BROWN - EVPI

34

The Expected Value of Perfect InformationDecision Large rise Small rise No change Small fall Large fall

Gold -100 100 200 300 0Bond 250 200 150 -100 -150

Stock 500 250 100 -200 -600

C/D 60 60 60 60 60

Probab. 0.2 0.3 0.3 0.1 0.1

If it were known with certainty that there will be a “Large Rise” in the mark

Large rise

... the optimal decision would be to invest in...

-100

250500 60

Stock

Similarly,…



TOM BROWN - EVPI

35

The Expected Value of Perfect InformationDecision Large rise Small rise No change Small fall Large fall

Gold -100 100 200 300 0Bond 250 200 150 -100 -150

Stock 500 250 100 -200 -600

C/D 60 60 60 60 60

Probab. 0.2 0.3 0.3 0.1 0.1

-100250

500 60

Expected Return with Perfect information =ERPI = 0.2(500)+0.3(250)+0.3(200)+0.1(300)+0.1(60) = $271Expected Return without additional information =Expected Return of the EV criterion = $130

EVPI = ERPI - EREV = $271 - $130 = $141



6.5 Bayesian Analysis - DecisionMaking with ImperfectInformation

• Bayesian Statistics play a role in assessing additional information

obtained from various sources.• This additional information may assist in refining original probability

estimates, and help improve decision making.

36



TOM BROWN – Using SampleInformation

• Tom can purchase econometric forecast results for $50.• The forecast predicts “negative” or “positive” econometric

growth.• Statistics regarding the forecast are:

37

The Forecast When the stock market showed a... predicted Large Rise Small Rise No Change Small Fall Large FallPositive econ. growth 80% 70% 50% 40% 0%Negative econ. growth 20% 30% 50% 60% 100%

When the stock market showed a large rise theForecast predicted a “positive growth” 80% of the time.

Should Tom purchase the Forecast ?



TOM BROWN – SolutionUsing Sample Information

• If the expected gain resulting from the decisionsmade with the forecast exceeds $50 , Tom shouldpurchase the forecast.

The expected gain =

Expected payoff with forecast – EREV• To find Expected payoff with forecast Tom should

determine what to do when:• The forecast is “positive growth”, • The forecast is “negative growth”. 38



TOM BROWN – SolutionUsing Sample Information

• Tom needs to know the following probabilities• P(Large rise | The forecast predicted “Positive”)• P(Small rise | The forecast predicted “Positive”)• P(No change | The forecast predicted “Positive ”)• P(Small fall | The forecast predicted “Positive”) • P(Large Fall | The forecast predicted “Positive”)• P(Large rise | The forecast predicted “Negative ”) •

P(Small rise | The forecast predicted “Negative”) • P(No change | The forecast predicted “Negative”) • P(Small fall | The forecast predicted “Negative”) • P(Large Fall) | The forecast predicted “Negative”) 39



TOM BROWN – SolutionBayes’ Theorem

• Bayes’ Theorem provides a procedure to calculate theseprobabilities

40

P(B|Ai)P(Ai)

P(B|A1)P(A1)+ P(B|A2)P(A2)+…+ P(B|An)P(An)P(Ai|B) =

Posterior ProbabilitiesProbabilities determinedafter the additional infobecomes available.

Prior probabilitiesProbability estimatesdetermined based oncurrent info, before the

new info becomes available.



TOM BROWN – Solution

Bayes’ Theorem

41

States of Prior Prob. Joint Posterior Nature Prob. (State|Positive) Prob. Prob.Large Rise 0.2 0.8 0.16 0.286

Small Rise 0.3 0.7 0.21 0.375

No Change 0.3 0.5 0.15 0.268

Small Fall 0.1 0.4 0.04 0.071Large Fall 0.1 0 0 0.000

X =

The Probability that the forecast is“positive”and the stock marketshows “Large Rise”.

• The tabular approach to calculating posteriorprobabilities for “positive” economical forecast




42


Small Rise 0.3 0.7 0.21 0.375

No Change 0.3 0.5 0.15 0.268


X =0.160.56

The probability that the stock marketshows “Large Rise”given that

the forecast is “positive”





43


Small Rise 0.3 0.7 0.21 0.375

No Change 0.3 0.5 0.15 0.268


X =

Observe the revision in

the prior probabilities

Probability(Forecast = positive) = .56





44

States of Prior Prob. Joint Posterior Nature Prob. (State|negative) Probab. Probab.Large Rise 0.2 0.2 0.04 0.091

Small Rise 0.3 0.3 0.09 0.205

No Change 0.3 0.5 0.15 0.341

Small Fall 0.1 0.6 0.06 0.136Large Fall 0.1 1 0.1 0.227

Probability(Forecast = negative) = .44

• The tabular approach to calculating posteriorprobabilities for “negative” economical forecast

oster or rev se



oster or rev seProbabilities

spreadsheet template

45

Bayesian Analysis

Indicator 1 Indicator 2

States Prior Conditional Joint Posterior States Prior Conditional Joint Posterior of Nature Probabilities Probabilities Probabilities Probabilites of Nature Probabilities Probabilities Probabilities ProbabilitesLarge Rise 0.2 0.8 0.16 0.286 Large Rise 0.2 0.2 0.04 0.091Small Rise 0.3 0.7 0.21 0.375 Small Rise 0.3 0.3 0.09 0.205No Change 0.3 0.5 0.15 0.268 No Change 0.3 0.5 0.15 0.341Small Fall 0.1 0.4 0.04 0.071 Small Fall 0.1 0.6 0.06 0.136Large Fall 0.1 0 0 0.000 Large Fall 0.1 1 0.1 0.227s6 0 0 0.000 s6 0 0 0.000s7 0 0 0.000 s7 0 0 0.000s8 0 0 0.000 s8 0 0 0.000

P(Indicator 1) 0.56 P(Indicator 2) 0.44

xpecte a ue o amp e



xpecte a ue o amp eInformation

EVSI• This is the expected gain from making decisions based on Sample

Information.• Revise the expected return for each decision using the posterior

probabilities as follows:

46



TOM BROWN – Conditional Expected Values

47

The revised probabilities payoff tableDecision Large rise Small rise No change Small fall Large fall

Gold -100 100 200 300 0

Bond 250 200 150 -100 -150

Stock 500 250 100 -200 -600

C/D 60 60 60 60 60

P(State|Positive) 0.286 0.375 0.268 0.071 0

P(State|negative) 0.091 0.205 0.341 0.136 0.227

EV(Invest in…….|“Positive” forecast) = =.286( )+.375( )+.268( )+.071( )+0( ) =

EV(Invest in …….| “Negative” forecast) =

=.091( )+.205( )+.341( )+.136( )+.227( ) =

-100 100 200 300 $840GOLD

-100 100 200 300 0

GOLD

$120



TOM BROWN – Conditional Expected Values• The revised expected values for each decision:

Positive forecast Negative forecast

EV(Gold|Positive) = 84 EV(Gold|Negative) =120

EV(Bond|Positive) = 180 EV(Bond|Negative) =65EV(Stock|Positive) = 250

EV(Stock|Negative) = -37

EV(C/D|Positive) = 60 EV(C/D|Negative) = 60

48

If the forecast is “Positive” Invest in Stock.

If the forecast is “Negative” Invest in Gold.



TOM BROWN – Conditional Expected Values

• Since the forecast is unknown before it is purchased, Tom can onlycalculate the expected return from purchasing it.

• Expected return when buying the forecast = ERSI = P(Forecast ispositive) (EV(Stock| Forecast is positive)) +P(Forecast is negative”) (EV(Gold| Forecast isnegative))= (.56)(250) + (.44)(120) = $192.5

49

E d V l f S li



Expected Value of SamplingInformation (EVSI)

• The expected gain from buying the forecast is:EVSI = ERSI – EREV = 192.5 – 130 = $62.5

• Tom should purchase the forecast. His expected gain is greater

than the forecast cost.• Efficiency = EVSI / EVPI = 63 / 141 = 0.45

50



TOM BROWN – Solution

EVSI spreadsheet template

51

Payoff Table

Large Rise Small Rise No Change Small Fall Large Fall s6 s7 s8 EV(prior) EV(ind. 1) EV(ind. 2)Gold -100 100 200 300 0 100 83.93 120.45Bond 250 200 150 -100 -150 130 179.46 67.05Stock 500 250 100 -200 -600 125 249.11 -32.95C/D Account 60 60 60 60 60 60 60.00 60.00d5d6d7d8Prior Prob. 0.2 0.3 0.3 0.1 0.1Ind. 1 Prob. 0.286 0.375 0.268 0.071 0.000 #### ### ## 0.56Ind 2. Prob. 0.091 0.205 0.341 0.136 0.227 #### ### ## 0.44Ind. 3 Prob.Ind 4 Prob.

RESULTSPrior Ind. 1 Ind. 2 Ind. 3 Ind. 4

optimal payoff 130.00 249.11 120.45 0.00 0.00optimal decision Bond Stock Gold

EVSI = 62.5EVPI = 141Efficiency= 0.44



6.6 Decision Trees• The Payoff Table approach is useful for a non-sequential or

single stage.

• Many real-world decision problems consists of a sequence ofdependent decisions.

• Decision Trees are useful in analyzing multi-stage decisionprocesses.

52

Ch i i f d i i



Characteristics of a decisiontree

• A Decision Tree is a chronological representation ofthe decision process.

• The tree is composed of nodes and branches.

53

A branch emanating from astate ofnature (chance) node corresponds to aparticular state of nature, and includesthe probability of this state of nature.

Decisionnode

Chancenode

P(S2)

P(S2)

A branch emanating from adecision node corresponds to adecision alternative. It includes acost or benefit value.



• BGD plans to do a commercial development on a property.• Relevant data

• Asking price for the property is 300,000 dollars.•

Construction cost is 500,000 dollars.• Selling price is approximated at 950,000 dollars.• Variance application costs 30,000 dollars in fees and expenses

• There is only 40% chance that the variance will be approved.• If BGD purchases the property and the variance is denied, the property can be sold

for a net return of 260,000 dollars.•

A three month option on the property costs 20,000 dollars, which will allow BGDto apply for the variance.

54

BILL GALLEN DEVELOPMENT COMPAN



BILL GALLEN DEVELOPMENTCOMPANY

• A consultant can be hired for 5000 dollars.• The consultant will provide an opinion about the approval of the

application• P (Consultant predicts approval | approval granted ) = 0.70• P (Consultant predicts denial | approval denied ) = 0.80

• BGD wishes to determine the optimal strategy• Hire/ not hire the consultant now,• Other decisions that follow sequentially.

55



• Construction of the Decision Tree• Initially the company faces a decision about hiring the consultant.

•

After this decision is made more decisions follow regarding• Application for the variance.• Purchasing the option.• Purchasing the property.

56

BILL GALLEN - Solution



BILL GALLEN - The Decision Tree

57

Buy land-300,000

Apply for variance

Apply for variance

-30,000

-30,000

03




58

12

-300,000 -500,000 950,000

Buy land Build Sell

-50,000

100,000

-70,000

260,000Sell

Build Sell950,000-500,000

120,000Buy land andapply for variance

-300000 – 30000 + 260000 =

-300000 – 30000 – 500000 + 950000 =

Purchase option andapply for variance




59

This is where we are at this stage

Let us consider the decision to hire a consultant



BILL GALLEN–

The Decision Tree

60

-5000

Apply for variance

Apply for variance

Apply for variance

Apply for variance

-5000

-30,000

-30,000

-30,000

-30,000

Let us consider thedecision to hire aconsultant

Done

Buy land-300,000

Buy land-300,000

BILL GALLEN Th D i i T




61

?

?

Build Sell950,000-500,000

260,000Sell

-75,000

115,000





62

?

?

Build Sell950,000-500,000

260,000Sell

-75,000

115,000

The consultant serves as a source for additional informationabout denial or approval of the variance.





63

?

?

Build Sell950,000-500,000

260,000Sell

-75,000

115,000

Therefore, at this point we need to calculate theposterior probabilities for the approval and denial

of the variance application





64

22

Build Sell950,000-500,000

260,000Sell

-75,00027

25115,000

23 24

26

The rest of the Decision Tree is built in a similar manner.

Posterior Probability of (approval |consultant predicts approval) = 0.70Posterior Probability of (denial |consultant predicts approval) = 0.30

?

?

.7

.3

Th D i i T



The Decision Tree Determining the Optimal Strategy

• Work backward from the end of each branch.

• At a state of nature node, calculate the expectedvalue of the node.

•

At a decision node, the branch that has thehighest ending node value represents the optimaldecision.

65




BILL GALLEN - The Decision TreeDetermining the Optimal Strategy

66

22

27

2523 24

26-75,000

115,000115,000

-75,000

115,000

-75,000

115,000

-75,000

115,000

-75,00022

115,000

-75,00058,000 ?

?0.30

0.70

Build Sell950,000-500,000

260,000Sell

-75,000

115,000

With 58,000 as the chance node value,we continue backward to evaluate

the previous nodes.




BILL GALLEN - The Decision TreeDetermining the Optimal Strategy

67

$10,000

$58,000

$-5,000

$20,000

$20,000Buy land; Applyfor variance

Build,Sell

Sellland

$-75,000

$115,000



BILL GALLEN - The Decision TreeExcel add-in: Tree Plan

68



BILL GALLEN - The Decision TreeExcel add-in: Tree Plan

69

6 7 Decision Making and



6.7 Decision Making andUtility

• Introduction• The expected value criterion may not be appropriate if the decision

is a one-time opportunity with substantial risks.• Decision makers do not always choose decisions based on the

expected value criterion.• A lottery ticket has a negative net expected return.• Insurance policies cost more than the present value of the expected

loss the insurance company pays to cover insured losses.

70



The Utility Approach

• It is assumed that a decision maker can rank decisions ina coherent manner.

• Utility values, U(V), reflect the decision maker’sperspective and attitude toward risk.

• Each payoff is assigned a utility value. Higher payoffs getlarger utility value.

• The optimal decision is the one that maximizes theexpected utility.

71



Determining Utility Values

• The technique provides an insightful look into the amount of riskthe decision maker is willing to take.

• The concept is based on the decision maker’s preference to taking

a sure payoff versus participating in a lottery.

72




Indifference approach for assigning utility values• List every possible payoff in the payoff table in ascending order.• Assign a utility of 0 to the lowest value and a value of 1 to the

highest value.• For all other possible payoffs (R ij) ask the decision maker the

following question:

73




Indifference approach for assigning utility values• Suppose you are given the option to select one of the following

two alternatives:• Receive $R ij (one of the payoff values) for sure,• Play a game of chance where you receive either

• The highest payoff of $R max with probability p, or• The lowest payoff of $R min with probability 1- p.

74




Indifference approach for assigning utility values

What value of p would make you indifferent

between the two situations?”

75

Rmin Rij

Rmax

p

1-p





The answer to this question is the indifferenceprobability for the payoff R ij and is used as theutility values of R ij.

76

Rmin Rij

Rmax

p

1-p





77

d1

d2

s 1 s 1

150

-50 140

100

Alternative 1A sure event

Alternative 2(Game-of-chance)

$100$150

-50p1-p

• For p = 1.0, you’llprefer Alternative 2.

• For p = 0.0, you’ll

prefer Alternative 1.• Thus, for some pbetween 0.0 and 1.0you’ll be indifferent

between the alternatives.

Example:





78

d1

d2

s 1 s 1

150

-50 140

100

Alternative 1A sure event

Alternative 2(Game-of-chance)

$100$150

-50p1-p

• Let’s assume the probability ofindifference is p = .7.

U(100)=.7U(150)+.3U(-50)= .7(1) + .3(0) = .7



TOM BROWN - Determining Utility Values• Data

• The highest payoff was $500. Lowest payoff was -$600.• The indifference probabilities provided by Tom are

• Tom wishes to determine his optimal investment Decision.

79

Payoff -600 -200 -150 -100 0 60 100 150 200 250 300 500 Prob. 0 0.25 0.3 0.36 0.5 0.6 0.65 0.7 0.75 0.85 0.9 1



TOM BROWN – Optimal decision (utility)

80

Utility Analysis Certain Payoff Utility-600 0

Large Rise Small Rise No Change Small Fall Large Fall EU -200 0.25Gold 0.36 0.65 0.75 0.9 0.5 0.632 -150 0.3Bond 0.85 0.75 0.7 0.36 0.3 0.671 -100 0.36Stock 1 0.85 0.65 0.25 0 0.675 0 0.5C/D Account 0.6 0.6 0.6 0.6 0.6 0.6 60 0.6d5 0 100 0.65d6 0 150 0.7d7 0 200 0.75d8 0 250 0.85Probability 0.2 0.3 0.3 0.1 0.1 300 0.9

500 1

RESULTSCriteria Decision ValueExp. Utility Stock 0.675



Three types of Decision Makers• Risk Averse -Prefers a certain outcome to a

chance outcome having the same expected value.

•

Risk Taking - Prefers a chance outcome to acertain outcome having the same expected value.

• Risk Neutral - Is indifferent between a chanceoutcome and a certain outcome having the sameexpected value.

81

decision maker.ppt

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