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Decision under uncertainty with limited information

Efstratios NikolaidisThe University of Toledo

December 2010

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Outline

1. Why decision-making is important2. Decision analysis: a powerful method to

make good decisions3. Solving decision problems4. Quantifying the value of information5. How to define and estimate probabilities

of one-time events 6. Lessons learnt

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1. Why decision-making is important

• We all make decisions that affect our personal and professional life.

• The welfare of companies and countries depends on the ability of their leaders to make good decisions.

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Example

Drill for oil• You own land in oil rich region• Can sell it for $18,500 or drill • Drilling costs $70,000• If no oil is found land becomes worthless• Sell or drill?

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Decision tree: Graphical representation of sequence of choices and uncertain eventsShows profit for each combination of choices and

outcomes of uncertain events.

0 -70000

($70,000)

$120,000 50000

$270,000 200000

$18,500.00 18500

Dril for Oil Decision

Drill

Sell

Dry

Wet

Soak

Decision

Uncertain event

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Both your decision and the outcomes of uncertain event affect

profit

Decision: Drill or sell?

Drill outcome Consequence (profit)

Drill No oil -$70,000 Drill Wet $50,000 Drill Soaking $200,000 Sell -- $18,500

Drill or sell?

Amount of oil

Consequence

Outcomes No oil Wet Soaking

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• In order to make a good decision you must consider:– Alternatives and their consequences

– Probabilities of outcomes of uncertain events – Your risk attitude (value of a sure amount of

$18,500 vs. opportunity of a windfall and risk of a big loss)

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Decisions in Product Development

• Objective: maximize profit (price, cost, performance)• Decision variables:

• Enterprise: price, volume • Engineering: configuration, material, dimensions

• Our decisions affect both performance and cost

Enterprise and engineering

design variables

Cost Performance

Demand Profit

Decision Consequence

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2. Decision analysis: a powerful method to make good decisions

• An approach that is based on common sense rules to make decisions in challenging situations.

• A good decision is based on careful consideration of alternatives, uncertainties and preferences.

• A good decision does not guarantee a good outcome, but it increases its likelihood, in the long run.

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Elements of a decision

• Values and objectives• Alternative courses of action• Model for predicting consequences• Uncertainty• Probabilities of outcomes of uncertain

events• Decision criterion

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Decision process

Structure decision•Model preferences•Model uncertainty•Develop predictive model for the consequences of each alternative course of action

Define and frame decision: understand the decision situation, examine values and preferences

Identify options

Choose best course of action

Assess sensitivity of decision to the assumptions

Appraise decision quality, assess value of additional information

Implement decision

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Reducing risk by considering more alternatives0 -80000

($70,000)

$120,000 40000

$270,000 190000

$18,500 8500

($10,000)

0 -80000

($70,000)

$120,000 40000

$270,000 190000

$18,500 8500

0 -80000

($70,000)

$120,000 40000

$270,000 190000

$18,500 8500

0 -70000

($70,000)

$120,000 50000

$270,000 200000

$18,500 18500

Test, drill or sell

Test

Drill

Sell

No structure

Open structure

Closed structure

Drill

Sell

Dry

W et

Soak

Drill

Sell

Dry

W et

Soak

Drill

Sell

Dry

W et

Soak

Dry

W et

Soak

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3. Solving decision problems• Decision rule: The best alternative is the one with the highest

expected monetary value

• Example, product development: Major household product manufacturer plans to produce innovative squeegee design– Commissioned vendor to analyze strength under heavy use. Vendor

reported that design is safe.– Company’s engineers reviewed report and are concerned that vendor

made an erroneous assumption. This has probability 0.25.– Development and marketing cost: $50 million– If vendor’s assumption is correct the company will make $100 million

revenue. Otherwise, revenue will be only $10 million.– Can make and test 10 prototypes. – Develop, test or cancel development?

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Quantifying accuracy of test

• P(pass test/valid assumptions)=0.98

• P(pass test/wrong assumptions)=0.1• Cost of test: $100,000

• Total probability theorem: P(pass test)=0.76

• Bayes’ rule: – P(valid assumptions/pass test)=0.9671– P(valid assumptions/fail test)=0.0625

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Decision tree

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Backward induction, first step: replace gambles with their expected profit

Expected profit

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Backward induction, second step: make decisions given what you know from the test

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Backward induction, third step: make final decision

Optimum strategy: First test product. Develop and market it if it passes test,

otherwise cancel development.

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4. Quantifying the value of information

• Principle: value of information from test = increase in expected profit from using the information from the test– Value of test=$35.75-$27.5=$8.25 million– Net value of test=value of test-cost=$8.15

million

• Before conducting a test estimate net value. Do not conduct a test with net negative value.

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Value of test with perfect information

• Perfect information: test would always predict correctly if assumptions were valid– Value of test=$37.5-$27.5=$10 million

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5. How to define and estimate probabilities of one-time events

• Objective probability=long term relative frequency – Example: P(heads in flip of fair coin)=0.5

– Long-term relative frequency– Objective measure

– Most people understand concept of objective probability

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Objective probability cannot quantify uncertainty in most

practical problems• There is little data• Too expensive to collect data• We cannot conduct a repeatable experiment for one-of-

a-kind events– A particular student will pass the Ph.D. qualifying exams in

spring 2011– Assumptions in a finite element analysis of a structure are

erroneous – Demand for Chevy Volt in US will not exceed 100,000 – Price of crude oil will exceed $150 per barrel by December 2012

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Subjective probability can help model uncertainty

• Principle 1: Probability is a decision maker’s (DM’s) belief that an outcome will materialize

• Principle 2: DM avoids a risky venture that will result in sure loss

• Belief leads to inclination to act. Elicit it by observing how DM makes choices in the face of uncertainty.

• Observe inclination to accept gambles in controlled experiments

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Objective vs. subjective probability

• Objective probability: how frequently does an earthquake stronger than 6 on the Richter scale occur in Southern California?

• Subjective probability: how likely is it that a proposition (demand for Chevy Volt<100,000) is true?

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Estimating subjective probability of a candidate winning 2008 U.S. presidential election by using

trading data

This ticket is worth $1 only if Mr. Obama

wins 2008 presidential election

Maximum buying price reflected a gambler’s belief that Mr. Obama would win election

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Trading data from

2008 U.S

. presidentialelection(http://new

sfutures.wordpress.com

)

40 50 60 70 80 90

100

2008-01-01

2008-01-15

2008-01-29

2008-02-12

2008-02-26

2008-03-11

2008-03-25

2008-04-07

2008-04-21

2008-05-05

2008-05-19

2008-06-02

2008-06-16

2008-06-30

2008-07-14

2008-07-28

2008-08-11

2008-08-25

2008-09-08

2008-09-22

2008-10-06

2008-10-20

2008-11-02

Da

te

Ticket Price (cents)

P(w

in)=0.5

P(w

in)=0.8

27DM is decisive to avoid sure loss

Ticket

Expert prefers ticket to sure amount.

Expert prefers sure amount to ticket.

Ticket price=$p ⇔ Subjective probability=p

$0 $1

Ticket

Sure amount

Sure amount

Sure amount

Eliciting expert’s probability

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Imprecision in probability elicitation

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Eliciting a decision maker’s probability distribution

• Estimating 5% percentile of water pump life if we cannot performtests

φ

Life (hrs)

CDF0.05

2000

Real life experiment Reference experiment: wheel of fortune

P(5% percentile ≤ 2000 hrs) = φ/3600

Ticket 1: Worth $1 only if life 5% percentile ≤ 2000 hrs

Ticket 1: Worth $1 only if needle settles in sector φ

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Combining judgments and data by using Bayes’ rule

Example– Judgment: 1 per 10 pumps fail on average– Posterior = likelihood × prior ×scaling constant

– Subjective probability converges to relative frequency and epistemic uncertainty decreases with amount of data

0 0.2 0.4 0.6 0.8 10

5

10

15

0

0.1

0.2

0.3

0.4

PriorPosteriorLikelihood

Failure Probability

PDF

Lik

elih

ood

0 0.2 0.4 0.6 0.8 10

10

20

30

0

0.05

0.1

0.15

PriorPosteriorLikelihood

Failure Probability

PDF

Lik

elih

ood

Data: 1 out of 20 pumps failed Data: 10 out of 200 pumps failed

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6. Lessons learnt

• Decision analysis is a power tool for making choices under uncertainty.

• It is based few on common sense rules.• How to make good decisions

– Know what you want– Make sure you are working on the right problem; ask your self if

you are addressing the right question– Consider many fundamentally different and creative alternatives.– Do not discount luck: carefully assess the likelihood of uncertain

events that affect the consequences of a decision.– Compare alternative courses of action on the basis of their

consequences – not the actions themselves.– Understand all important issues in a decision and include them

in your model

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• In most practical decisions we do not have enough data to estimate relative frequencies. Objective probability is inadequate for modeling uncertainty.

• Subjective probability enables decision-maker to model uncertainty on the basis of both judgment and data.

• Subjective probability has a solid theoretical justification derived from first principles.

• Can combine judgment with data by using Bayes’ rule. Subjective probability converges to relative frequency with amount of data increasing.

• Ambiguity aversion leads to indecision. Some people’s behavior is at odds with the precepts of subjective probability.