Week 07-Decision Analysis Ch19

7/23/2019 Week 07-Decision Analysis Ch19

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1 Introduction to Decision Theory

2 Decision Making Using Posterior (or

revised) Probabilities


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States of nature: A set of potential future conditions

that affects decision results

Alternatives: A set of alternative actions for thedecision maker to choose from

Payoffs: A set ofpayoffs for each alternative under

each potential state of nature


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Maximin: Identify the minimum (or worst) possible payoff for

each alternative and select the alternative that maximizes

the worst possible payoff (Pessimistic)

Maximax: Identify the maximum (or best) possible payoff for

each alternative and select the alternative that maximizes

the best possible payoff (Optimistic)

Expected monetary value (EMV) criterion: Usingpriorprobabilities for the states of nature, compute the expected

payoff for each alternative and select the alternative with the

largest expected payoff


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A Payoff Table for the Condominium Complex Situation


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1. If a small complex is built, the worst possible payoffis $8 million

2. If a medium complex is built, the worst possiblepayoff is $5 million

3. If a large complex is built, the worst possible payoff

is -$11 millionSince the maximumofthese worst possible payoffs is$8 million, the developer should choose to build asmall complex


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1. If a small complex is built, the best possiblepayoff is $8 million

2. If a medium complex is built, the best possiblepayoff is $15 million

3. If a large complex is built, the best possiblepayoff is $22 million

Since the maximum of these best possible payoffs is$22 million, the developer should choose to build alarge complex.


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If prior probabilities are 0.3 for low demand and 0.7 for

high demand there is risk of probability errors

1. Small complex:Expected value = 0.3($8 million) + 0.7($8 million) = $8 million

2. Medium complex:

Expected value = 0.3($5 million) + 0.7($15 million) = $12 million

3. Large complex:

Expected value = 0.3(-$11 million)+.7($22 million)=$12.1 million

Choose large complex


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Sometimes it is possible to determine exactly which state ofnature will occur in the future, e.g., by obtaining perfectinformation

EVPI = Expected value under certainty expected valueunder risk

To calculate EVcertainty : find the highest payoff for each stateof nature ($8 mil for low and $22 mil for high demand) and

then:EVcertainty = 0.3 * $8 + 0.7 * $22 = $17.8 million

EVPI = $17.8 - $12.1 = $5.7 million


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A Payoff Table for the capacity expansion plan for a companyis given below

Possible Future Demand

Alternatives Low Moderate HighSmall Facilities $10 $10 $10

Medium Facilities $7 $12 $12Large Facilities -$4 $2 $16

1. Find the best alternative (and the resulting payoff) if it is knownwith certainty that demand will be moderate.

2. Find the alternative using maximin and maximax criterion3. Construct a decision tree assuming prior probability of low,

moderate, and high demand are 0.3, 0.5, and 0.2

4. Find the expected monetary value (EMV) for each alternative(small, medium, large). What is the best alternative? Find

EVPI.


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1. Medium facility, $12 million.

2.

Minimum payoffs:

Large = $4 M, Medium = $7M, Small = $10M

Maximum payoff of the minimums (maximin): Smallfacility at $10M

Maximum payoff :

Small = $10M, Medium = $12M, Large = $16M

Maximax payoff: Large facility at $16M


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3. Dec. Tree

4.

EMV (Small) = (10)(.3) + (10)(.5) + (10)(.2) = $10MEMV(Medium) = (7)(.3) + (12)(.5) + (12)(.2) = $10.5M

EMV(Large) = (4)(.3) + (2)(.5) + (16)(.2) = $3M

Best alternative: Medium facility


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4.

EVcertainty = (10)(.3) + (12)(.5) + (16)(.2) = $12.2M

EVPI = $12.2M - $10.5M = $1.7M

Possible Future Demand

Alternatives Low Moderate HighSmall Facilities $10 $10 $10

Medium Facilities $7 $12 $12Large Facilities -$4 $2 $16


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When we use expected value to

choose the best alternative, we call

this prior decision analysis

Often, sample information can be

obtained to help us make a better

decision

In this case, we compute expected

values by using posterior probabilities

We call this posterior decision analysis

New

Information

RevisedProbability

Prior

Probability


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The probability of an event A, given that

the event B has occurred:

P(A|B) = P(AB) / P(B)

= P(A) P(B|A) / P(B)


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An oil company trying to decide about do drilling

or do not.

There are three states of nature

1. No oil (S1) P(None) = 0.7

2. Some oil (S2) P(Some) = 0.2

3. Much oil (S3) P(Much) = 0.1


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State of Nature

Alternatives No Oil (0.7) Some Oil (0.2) Much Oil (0.1)

Drill -$700,000 $500,000 $2,000,000

Do not drill $0 $0 $0

EV(drill) = 0.7x(-$700,000) + 0.2x$500,000 + 0.1x$2,000,000 = -$190,000

EV(Do not drill) = $0Prior Analysis tells us Do not drill.


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The Oil Drilling Case - Summary

Prior Analysis tells us Do not drill. However, there is

possibility for Seismic Experiment . Seismic Experiment hasthree readings: low, medium and high

Info of posterior probability (e.g. from historical data):

P(low|no oil)=0.91 P(medium|no oil)=0.05 P(high|no oil)=0.04

P(low|some oil)=0.04 P(medium|some oil)=0.94 P(high|some oil)=0.02

P(low|much oil)=0.01 P(medium|much oil)=0.03 P(high|much oil)=0.96

The questions:

1) Do seismic survey (cost $100,000) or not

2) If the seismic result=low, Drill or Do not drill?

If the seismic result=medium, Drill or Do not drill?

If the seismic result=high, Drill or Do not drill?


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128.0

)96.0)(1.0()02.0)(2.0()04.0)(7.0(

)|()|()()|()(

)()()()(

75000.0128.0

)96.0(1.0

)(

)|()()|(

03125.0128.0

)02.0(2.0)(

)|()()|(

21875.0128.0

)04.0(7.0

)(

)|()()|(

muchhighPsomehighPsomePnonehighPnoneP

highmuchPhighsomePhighnonePhighP

highP

muchhighPmuchPhighmuchP

highPsomehighPsomePhighsomeP

highP

nonehighPnonePhighnoneP

Previously :

P(none) = 0.7, P(some) = 0.2, P(much) = 0.1

Posterior probability of high reading:

P(high|none) = 0.04, P(high|some) = 0.02, P(high|much) = 0.96


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Expected payoff ofsampling

Expected payoff ofno sampling


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Expected payoff of sampling

EPS = .646($0) + 0.226($334,061) + 0.128($1,362,500) =

$249,898

Expected payoff of no sampling = 0

Expected value of sample information

EVSI = EPS EPNS

EVSI = $249,898 - $0 = $249,898

Expected net gain of sampling

ENGS = EVSI - $100,000

ENGS = $249,898 - $100,000 = $149,898


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An investor wishes to choose between investing money in (1 ) a high-risk

stock, (2) a low-risk stock, or (3) a savings account. The payoffs receiveddepend on the behavior of the stock market - whether the market goes up,stays the same, or goes down over the investment period.

The investor can hire an economist who will predict the future marketbehavior. The results of the consultation: (1) "economist says up (2)

"economist says flat" (the same), or (3) economist says down. See thefollowing table

Investment Decision

True Market State

Economist's Prediction Up Flat Down

Economist says up 0.80 0.15 0.20Economist says flat 0.10 0.70 0.20Economist says down 0.10 0.15 0.60

In this table P(economist says up | market up) = 0.80. Prior

probabilities of market state up, flat, and down = 0.5, 0.3, and 0.2.


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The next figure gives an incompletedecision tree for the investor's situation.

1. a. Decision Alternatives?

b. States of Nature?

c. Sampling information?

2. Carry out a prior analysis of theinvestor's decision problem and

determine the best investment choice3. Set up probability revision tables to

find:

Probability economist says up,

Posterior probabilities market up,

market flat, and market down giventhat the "economist says up."

Repeat for Probability economist

says flat and economist says down

Fill the decision tree


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4. Carry out a posterior analysis of the investor's decisionproblem. That is, determine the best investment choicethat should be made

5. Calculate the EPS, the EPNS and the EVSI


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1. a. Alternatives: Savings Account, High-Risk stock, Low-Risk stock.

b. States of Nature: Market up, Market flat, Market down

c. Sampling info: says up, says down, says flat

2. EV(HighRisk) = (1500)(.5) + (100)(.3) + (1000)(.2) = 580

EV(LowRisk) = (1000)(.5) + (200)(.3) + (100)(.2) = 540

EV(SavingsAcct) = (500)(.5) + (500)(.3) + (500)(.2) = 500

Choose HighRisk stock; payoff = 580

3.


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3.

a. P(economist says up) = (.5)(.8) + (.15)(.3) + (.2)(.2) = .485

Posterior Probabilities: .8247, .0928, .0825b. P(economist says flat) = (.5)(.1) + (.7)(.3) + (.2)(.2) = .300

Posterior Probabilities: .1667, .7000, .1333

c. P(economist says flat) = (.5)(.1) + (.15)(.3) + (.6)(.2) = .215

Posterior Probabilities: .2326, .2093, .5581


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4.

a. Economist says: Market Up

EMV(HighRisk) = (1500)(.8247) + (100)(.0928) + (1000)(.0825) = 1163.83

EMV(LowRisk) = (1000)(.8247) + (200)(.0928) + (100)(.0825) = 835.05EMV(SavingsAcct) = (500)(.8247) + (500)(.0928) + (500)(.0825) = 500

Choose High-Risk stock

b. Economist says: Market Flat

EMV(HighRisk) = (1500)(.1667) + (100)(.7) + (1000)(.1333) = 186.75

EMV(LowRisk) = (1000)(.1667) + (200)(.7) + (100)(.1333) = 293.37

EMV(SavingsAcct) = 500

Choose Savings Acct

c. Economist says: Market Down

EMV(HighRisk) = (1500)(.2326) + (100)(.2093) + (1000)(.5581) =188.27

EMV(LowRisk) = (1000)(.2326) + (200)(.2093) + (100)(.5581) = 218.65

EMV(SavingsAcct) = 500

Choose Savings Acct


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4.

a. EPS = (.485)(1163.92) + (.3)(500) + (.215)(500) = 821.96

b. EPNS = 580

c. EVSI = 821.96 580 = 241.96

d. Max amount to pay economist for advice: 242.00

Week 07-Decision Analysis Ch19

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