1 Decision Analysis - Part 2 Aslı Sencer Graduate Program in Business Information Systems.

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Decision Analysis - Part 2 Decision Analysis - Part 2

Aslı SencerAslı Sencer

Graduate Program in Business Information Systems

BIS 517- Aslı SencerBIS 517- Aslı Sencer 22

Ex: Decision to buy insuranceEx: Decision to buy insurance

EventEvent ProbabilityProbability

ACT (choice)ACT (choice)

Buy insuranceBuy insuranceDo not buy Do not buy insuranceinsurance

FireFire 0.0020.002 -$100-$100 -$40,000-$40,000

No fireNo fire 0.9980.998 -$100-$100 00

Expected PayoffExpected Payoff -$100-$100 -$80-$80

Best act

IS IT SURPRISING?


Is Bayes decision rule invalid?Is Bayes decision rule invalid?

No, actually the true worth of outcomes is No, actually the true worth of outcomes is not completely reflected by the payoffs!not completely reflected by the payoffs!

Two approaches:Two approaches: Certainty EquivalentsCertainty Equivalents Utility FunctionUtility Function

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Decision Making Using Decision Making Using Certainty EquivalentsCertainty Equivalents

The The certainty equivalentcertainty equivalent (CE) is the payoff (CE) is the payoff amount we would accept in lieu of under-amount we would accept in lieu of under- going the uncertain situation.going the uncertain situation.

Shirley Smart would pay $25 to insure her 1983Shirley Smart would pay $25 to insure her 1983 Toyota against total theft loss. CE = – $25.Toyota against total theft loss. CE = – $25.

For $1,000, Willy B. Rich would sell his Far-For $1,000, Willy B. Rich would sell his Far- Fetched Lottery rights. CE = $1,000.Fetched Lottery rights. CE = $1,000. Game: win $5000 with probability 50% Game: win $5000 with probability 50%

win 0 with probability 50%win 0 with probability 50%


Risk PremiumsRisk Premiums

A situation’s A situation’s risk premiumrisk premium (RP) is the (RP) is the difference between its expected payoff (EP) difference between its expected payoff (EP) and certainty equivalent (CE):and certainty equivalent (CE):

RP = EP RP = EP CE CE Shirley Smart’s car is worth $1,000 and there Shirley Smart’s car is worth $1,000 and there

is a 1% chance of its being stolen. Thus, is a 1% chance of its being stolen. Thus, going without insurance hasgoing without insurance has

EP = (– $1,000)(.01) + ($0)(.99) = – $10EP = (– $1,000)(.01) + ($0)(.99) = – $10RP = – $10 – (– $25) = $15RP = – $10 – (– $25) = $15


Risk PremiumsRisk Premiums

Playing the Far-Fetched Lottery has EP = $2,500.Playing the Far-Fetched Lottery has EP = $2,500. Thus,Thus, For Willy B. Rich, For Willy B. Rich,

RP = EP – CE = $2,500 – ($1,000) = $1,500RP = EP – CE = $2,500 – ($1,000) = $1,500 For Lucky Chance,For Lucky Chance,

RP = EP – CE = $2,500 – (– $100) = $2,600RP = EP – CE = $2,500 – (– $100) = $2,600

Different people will have different CEs and RPs Different people will have different CEs and RPs for the same circumstance.for the same circumstance. They have different They have different attitudes toward riskattitudes toward risk..


Attitude Toward RiskAttitude Toward Risk

People with positive RPs are People with positive RPs are risk avertersrisk averters.. Lucky Chance has greater risk aversion than Willy B. Lucky Chance has greater risk aversion than Willy B.

Rich, as reflected by her greater RP.Rich, as reflected by her greater RP. We cannot compare Shirley’s risk aversion to the others’ We cannot compare Shirley’s risk aversion to the others’

because circumstances differ.because circumstances differ.

Risk averse persons have RPs that increase:Risk averse persons have RPs that increase: When the downside amounts become greater.When the downside amounts become greater. Or when the chance of downside increases.Or when the chance of downside increases.

A A risk seekerrisk seeker will have negative RP. will have negative RP. A A risk neutralrisk neutral person has zero RP. person has zero RP.


Maximizing Certainty EquivalentMaximizing Certainty Equivalent

A plausible axiom:A plausible axiom:Decision makers will prefer the act yielding Decision makers will prefer the act yielding greatest certainty equivalent.greatest certainty equivalent.

A logical conclusion:A logical conclusion:The ideal decision criterion is to maximizeThe ideal decision criterion is to maximizecertainty equivalent.certainty equivalent. Doing so guarantees taking the Doing so guarantees taking the preferred actionpreferred action..

But But CEs are difficult to determine.CEs are difficult to determine. One approach One approach is to is to discount the EPsdiscount the EPs.. RP = EP – CE implies that CE = EP – RP.RP = EP – CE implies that CE = EP – RP.


Using Risk Premiums to Get Using Risk Premiums to Get Certainty EquivalentsCertainty Equivalents

Ponderosa Records president has the following risk premiums, Ponderosa Records president has the following risk premiums, keyed to the downside.keyed to the downside.

These were found by extrapolating from three equivalencies These were found by extrapolating from three equivalencies (white boxes).(white boxes). Exact amounts are unknowable, but these values seem to fit his risk Exact amounts are unknowable, but these values seem to fit his risk

profile.profile.

BIS 517- Aslı SencerBIS 517- Aslı Sencer

Calculation of Risk Premiums for Calculation of Risk Premiums for PanderosaPanderosa

The president of panderosa Record Co. is asked the The president of panderosa Record Co. is asked the following:following:

How much would you be willing to pay for insuring How much would you be willing to pay for insuring $100.000 recording equipment if there is 1% chance of $100.000 recording equipment if there is 1% chance of losing them due to external occasions. Note that, herelosing them due to external occasions. Note that, here

Expected payoff=$0(0.99)+(-$100.000)(0.01)=-$1000Expected payoff=$0(0.99)+(-$100.000)(0.01)=-$1000 He is willing to pay $2500 to get rid of this danger. He is willing to pay $2500 to get rid of this danger.

Certainty equivalent= -$2500Certainty equivalent= -$2500

RP=-1000-(-2500)=$1500RP=-1000-(-2500)=$1500

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Decision Tree Analysis with CEs Decision Tree Analysis with CEs (Discounted Expected Payoffs)(Discounted Expected Payoffs)


How Good is the Analysis?How Good is the Analysis?

This result is different from that of ordinary back folding This result is different from that of ordinary back folding (Bayes decision rule).(Bayes decision rule). It specifically reflects underlying risk aversion.It specifically reflects underlying risk aversion. The result must be correct if CEs are right.The result must be correct if CEs are right.

The major weakness is the ad hoc manner for getting the The major weakness is the ad hoc manner for getting the RPs, and hence the CEs.RPs, and hence the CEs. Many assumptions are made in extrapolating to get the table of Many assumptions are made in extrapolating to get the table of

RPs.RPs.

There is a cleaner way to achieve the same thing using There is a cleaner way to achieve the same thing using utilitiesutilities. .


Decision Making with UtilityDecision Making with Utility

Expected monitary value may not accurately Expected monitary value may not accurately reflect the DM’s preference when significant reflect the DM’s preference when significant

risks are involved!risks are involved!

It is also hard to evaluate risk premiums to It is also hard to evaluate risk premiums to calculate certainty equivalents.calculate certainty equivalents.

An alternative aproach is to replace payoffs with An alternative aproach is to replace payoffs with utilitiesutilities..

Max. certainty equivalentMax. certainty equivalent Max. utility Max. utility


Utility Utility AssumptionsAssumptions

Consider a set of outcomes, Consider a set of outcomes, OO11, , OO22, ..., , ..., OOnn. The . The

following assumptions are made:following assumptions are made: Preference rankingPreference ranking can be done. can be done. Transitivity of preferenceTransitivity of preference: A is preferred to B and B to : A is preferred to B and B to

C, then A must be preferred to C.C, then A must be preferred to C. ContinuityContinuity: Consider : Consider OObetweenbetween. . Take a gamble between Take a gamble between

two more extreme outcomes; winning yields two more extreme outcomes; winning yields OObestbest and and

losinglosing O Oworstworst.. There is a win probability There is a win probability qq making you making you

indifferent betweenindifferent between getting getting OObetweenbetween and gambling. Such a and gambling. Such a

gamble is called a gamble is called a reference lotteryreference lottery. .


Utility AssumptionsUtility Assumptions

ContinuityContinuity (continued): (continued): e.g., +$1,000 v. Far-Fetched Lottery, you pick e.g., +$1,000 v. Far-Fetched Lottery, you pick qq.. For Willy B. Rich, For Willy B. Rich, qq = .5. (His CE was = +$1,000.) = .5. (His CE was = +$1,000.) For Lucky Chance, For Lucky Chance, qq = .9. = .9. If the win probability were .99, would you risk +$1,000 to If the win probability were .99, would you risk +$1,000 to

gamble? What is your gamble? What is your qq??

SubstitutabilitySubstitutability: In a decision structure, you would : In a decision structure, you would willingly substitute for any outcome a gamble equally willingly substitute for any outcome a gamble equally preferred.preferred. One outcome on Lucky Chance’s tree is +$1,000; she would One outcome on Lucky Chance’s tree is +$1,000; she would

accept substituting for it the Far-Fetched Lottery gamble with .9 accept substituting for it the Far-Fetched Lottery gamble with .9 win probability.win probability.


Utility Assumptions and ValuesUtility Assumptions and Values

Increasing preferenceIncreasing preference: Raising : Raising qq makes any makes any reference lottery more preferred.reference lottery more preferred. Anybody would prefer the revised Far-Fetched Lottery when Anybody would prefer the revised Far-Fetched Lottery when

two coins are tossed and just one head will win the $10,000. two coins are tossed and just one head will win the $10,000. (The win probability goes from .5 to .75.) You still might not like (The win probability goes from .5 to .75.) You still might not like that gamble! that gamble!

Outcomes can be assigned Outcomes can be assigned utility valuesutility values arbitrarilyarbitrarily, so , so that the more preferred always gets the greater value:that the more preferred always gets the greater value:

uu((OObestbest) = 10 ) = 10 uu((OOworstworst)=0 )=0 uu((OObetweenbetween)=5)=5 Willy has Willy has uu(+$10,000) = 500, (+$10,000) = 500, uu$5,000) = 0 and $5,000) = 0 and uu(+$1,000) = (+$1,000) =

250. These are 250. These are hishis values only. values only.


Utility ValuesUtility Values

Lucky has different values:Lucky has different values: uu(+$10,000) = 50,(+$10,000) = 50,uu$5,000) = $5,000) = 99, and 99, and uu(+$1,000) = 35.1.(+$1,000) = 35.1.

Like temperature, where 0Like temperature, where 0oo and 100 and 100oo are different states are different states on Celsius and Fahrenheit scales, so on Celsius and Fahrenheit scales, so utility scales may utility scales may differdiffer..

The freezing point for water is 0The freezing point for water is 0oo C and the boiling point C and the boiling point 100100oo C. In-between states will have values in that C. In-between states will have values in that range, and hotter days will have greater temperature range, and hotter days will have greater temperature values than cooler.values than cooler.

So, too, with utility values. So, too, with utility values. They will fall into the range They will fall into the range defined by the extreme outcomes, defined by the extreme outcomes, OOworstworst and and OObestbest. . More More preferred outcomes will have greater utilities preferred outcomes will have greater utilities



A reference lottery can be used to find the utility A reference lottery can be used to find the utility for an outcome for an outcome OObetweenbetween by: by: FirstFirst, establish an indifference win probability , establish an indifference win probability qqbetweenbetween

making it equally preferred to the gamble:making it equally preferred to the gamble: OObestbest with probability with probability qqbetweenbetween and and

OOworstworst with probability 1 with probability 1 qqbetweenbetween SecondSecond, compute the lottery’s expected utility:, compute the lottery’s expected utility:

uu((OObetweenbetween))==uu((OObestbest)()(qqbetweenbetween) + ) + uu((OOworstworst)(1 )(1 qqbetweenbetween) )



Using the Far-Fetched Lottery as reference:Using the Far-Fetched Lottery as reference:

The indifference The indifference qq plays a role analogous to the plays a role analogous to the thermometer, reading thermometer, reading attitudeattitude towards the outcome towards the outcome similarly to measuring temperature.similarly to measuring temperature.

LotteryLottery

OutcomesOutcomes

WillyWilly LuckyLucky

Prob.Prob. UtilityUtility Prob.Prob. UtilityUtility

OObestbest ((+$10,000)+$10,000) qq=.5=.5 500500 qq=.9=.9 5050

OOworstworst ( ($5,000)$5,000) 1 1 .5.5 00 1 1 .9.9 9999

Expected Utility:Expected Utility: 250250 35.135.1

OObetweenbetween (+$1,000): (+$1,000): 250250 35.135.1


The Utility FunctionThe Utility Function

Utility values assigned to monetary outcomes constitute a Utility values assigned to monetary outcomes constitute a utility functionutility function..

From a few points From a few points we may graph the utility functionwe may graph the utility function and and apply it over a monetary range.apply it over a monetary range.

Those points may be obtained Those points may be obtained from an interview posing from an interview posing hypothetical gambleshypothetical gambles.. Using u(+$10,000)=100 and uUsing u(+$10,000)=100 and u$5,000)=0 Shirley Smart gave the $5,000)=0 Shirley Smart gave the

following equivalencies:following equivalencies: A: +$10,000 A: +$10,000 @@ qqA A v v $5,000 $5,000 ≡ +$1,000 if ≡ +$1,000 if qqA A =.70=.70 B: +$10,000 B: +$10,000 @@ qqB B v +$1,000 ≡ +$5,000 if v +$1,000 ≡ +$5,000 if qqB B =.75=.75 C1: +$1,000 C1: +$1,000 @@ qqC1C1 v v $5,000 ≡ $5,000 ≡ $500 if $500 if qqC1C1 =.70=.70 C2: +$1,000 C2: +$1,000 @@ qqC2C2 v v $5,000 ≡ $5,000 ≡ $2,000 if $2,000 if qqC2C2 =.30=.30


Shirley’s Utility FunctionShirley’s Utility Function

Shirley’s utilities for the equivalent amounts are equal to the Shirley’s utilities for the equivalent amounts are equal to the respective expected utilities:respective expected utilities: uu(+$1,000) = (+$1,000) = uu(+$10,000)(.70) + (+$10,000)(.70) + uu$1,000)(1$1,000)(1.70).70)

= 100(.70) + 0(1= 100(.70) + 0(1.70) = 70.70) = 70 uu(+$5,000) = (+$5,000) = uu(+$10,000)(.75) + (+$10,000)(.75) + uu$1,000)(1$1,000)(1.75).75)

= 100(.75) + 70(1 = 100(.75) + 70(1 .75) = 92.5 .75) = 92.5 uu$500) = $500) = uu(+$1,000)(.70) + (+$1,000)(.70) + uu$5,000)(1$5,000)(1.70).70)

= 70(.70) + 0(1 = 70(.70) + 0(1 .70) = 49 .70) = 49 uu(($2,000) = $2,000) = uu(+$1,000)(.30) + (+$1,000)(.30) + uu$5,000)(1$5,000)(1.30).30)

= 70(.30) + 0(1 = 70(.30) + 0(1 .30) = 21 .30) = 21 Altogether, Shirley gave 6 points, plotted on the following graph. Altogether, Shirley gave 6 points, plotted on the following graph.

The smoothed curve fitting through them defines her utility function. The smoothed curve fitting through them defines her utility function.


Shirley’s Utility FunctionShirley’s Utility Function


Using the Utility FunctionUsing the Utility Function

This utility function applies to the Ponderosa This utility function applies to the Ponderosa decision.decision.


Using the Utility FunctionUsing the Utility Function

Read the utility payoffsRead the utility payoffs corresponding to the net monetary corresponding to the net monetary payoffs.payoffs.

Apply the Bayes decision rule, with either:Apply the Bayes decision rule, with either: A utility payoff table, computing the expected payoff each act.A utility payoff table, computing the expected payoff each act. Or a decision tree, folding it back.Or a decision tree, folding it back.

The The certainty equivalentcertainty equivalent amount for any act or node amount for any act or node may may be found from the expected utility by reading the curve in be found from the expected utility by reading the curve in reverse.reverse.

The following Ponderosa Records decision tree was folded The following Ponderosa Records decision tree was folded back using utility payoffs.back using utility payoffs.


Decision Tree AnalysisDecision Tree Analysiswith Utilitieswith Utilities


Shape of Utility Curve and Shape of Utility Curve and Attitude Toward RiskAttitude Toward Risk

The following shapes generally apply.The following shapes generally apply.

The The risk averterrisk averter has has decreasing marginal util- ilitydecreasing marginal util- ility for money. He for money. He will buy casualty insurance and losses weigh more heavily than like will buy casualty insurance and losses weigh more heavily than like gains.gains.

Risk seekersRisk seekers like some unfavorable gambles. like some unfavorable gambles. Risk neutralityRisk neutrality values money at its face amount. values money at its face amount.


Important Utility RamificationsImportant Utility Ramifications

Hybrid shapes (like Shirley’s) imply Hybrid shapes (like Shirley’s) imply shifting attitudesshifting attitudes as monetary as monetary ranges change. ranges change.

Regardless of shape, Regardless of shape, maximizing expected utility also maximizing expected utility also maximizes certainty equivalent.maximizes certainty equivalent. Therefore, applying Bayes decision rule with utility payoffs Therefore, applying Bayes decision rule with utility payoffs

discloses the discloses the preferred actionpreferred action..

Primary impediments to implementation:Primary impediments to implementation: Clumsiness of the interview process.Clumsiness of the interview process. Multiple decision makers.Multiple decision makers. Attitudes change with circumstances and time.Attitudes change with circumstances and time.


Ratification ofRatification ofBayes Decision RuleBayes Decision Rule

Over narrow monetary ranges, utility curves Over narrow monetary ranges, utility curves resemble resemble straight linesstraight lines..

For a straight line, expected utility equals the utility For a straight line, expected utility equals the utility of the expected monetary payoff.of the expected monetary payoff.

Maximizing expected monetary payoff then also Maximizing expected monetary payoff then also maximizes expected utility.maximizes expected utility. Thus: Thus: The Bayes decision rule discloses the preferred action The Bayes decision rule discloses the preferred action

as long as the outcomes are not extreme.as long as the outcomes are not extreme. Managers can then Managers can then delegatedelegate decision making decision making

without having to find utilities. without having to find utilities. Preferred actions will Preferred actions will be found by the staff.be found by the staff.


Using Utility Functions with Using Utility Functions with PrecisionTreePrecisionTree

PrecisionTree can be used to evaluate decision trees with with exponential and logarithmic utility functions.

To get started, click on the name box of a decision tree and the Tree Setting dialog box appears, as shown next.

12

A B1

0tree #1

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Tree Settings Dialog BoxTree Settings Dialog Box (Figure 6-14)(Figure 6-14)

1. Check the Use Utility Function box.

1. Check the Use Utility Function box.

2. Select the type of utility function in the Function line. Here exponential is chosen.

2. Select the type of utility function in the Function line. Here exponential is chosen.

3. Select the risk coefficient, R, in the R value line. Here 10,000 is used.

3. Select the risk coefficient, R, in the R value line. Here 10,000 is used.

4. Select Expected Utility in the Display line. Other options are Certainty Equivalent and Expected Value.

4. Select Expected Utility in the Display line. Other options are Certainty Equivalent and Expected Value.

5. Click OK.5. Click OK.

3131

Decision Tree with Exponential Utility Function Decision Tree with Exponential Utility Function

for R = 10,000for R = 10,000 (Figure 6-15)(Figure 6-15) 123456789

10111213141516171819202122232425262728293031323334

A B C D E F80.0% 0.0

$90,000 1FALSE

-$50,000 -4820.0% 0.0

$0 -24450.0%

$10,000 -1TRUE 0

$0 -1FALSE

-$15,000 -220.0% 0.0

$90,000 1FALSE

-$50,000 -53180.0% 0.0

$0 -66450.0%

$0 -3TRUE 0

$0 -3

050.0% 0

$100,000 1FALSE

-$60,000 -20150.0% 0

$0 -402TRUE

$0 0TRUE 1

$0 0

Ponderosa Record Company

Test market

Don't test market

Favorable

Unfavorable

Market nationally

Abort

Success

Failure

Market nationally

Abort

Success

Failure

Market nationally

Abort

Success

Failure

The optimal strategy is:

1. Not test market and to abort.

2. The corresponding expected utility is 0.

The optimal strategy is:

1. Not test market and to abort.

2. The corresponding expected utility is 0.


Utility FunctionsUtility Functions

R: The risk tolerence

1 Decision Analysis - Part 2 Aslı Sencer Graduate Program in Business Information Systems.

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