EMGT 501 HW Solutions Chapter 12 - SELF TEST 9 Chapter 12 - SELF TEST 18

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© 2005 Thomson/South-Western© 2005 Thomson/South-Western

EMGT 501

HW Solutions

Chapter 12 - SELF TEST 9

Chapter 12 - SELF TEST 18

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Chapter 14Chapter 14Decision AnalysisDecision Analysis

Problem FormulationProblem Formulation Decision Making without ProbabilitiesDecision Making without Probabilities Decision Making with ProbabilitiesDecision Making with Probabilities Risk Analysis and Sensitivity AnalysisRisk Analysis and Sensitivity Analysis Decision Analysis with Sample Decision Analysis with Sample

InformationInformation Computing Branch ProbabilitiesComputing Branch Probabilities

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Problem FormulationProblem Formulation

A decision problem is characterized by A decision problem is characterized by decision alternatives, states of nature, and decision alternatives, states of nature, and resulting payoffs.resulting payoffs.

The The decision alternativesdecision alternatives are the different are the different possible strategies the decision maker can possible strategies the decision maker can employ.employ.

The The states of naturestates of nature refer to future events, refer to future events, not under the control of the decision not under the control of the decision maker, which may occur. States of nature maker, which may occur. States of nature should be defined so that they are mutually should be defined so that they are mutually exclusive and collectively exhaustive.exclusive and collectively exhaustive.

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Influence DiagramsInfluence Diagrams

An An influence diagraminfluence diagram is a graphical is a graphical device showing the relationships among device showing the relationships among the decisions, the chance events, and the decisions, the chance events, and the consequences.the consequences.

Squares or rectanglesSquares or rectangles depict decision depict decision nodes.nodes.

Circles or ovalsCircles or ovals depict chance nodes. depict chance nodes. DiamondsDiamonds depict consequence nodes. depict consequence nodes. Lines or arcsLines or arcs connecting the nodes show connecting the nodes show

the direction of influence.the direction of influence.

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Payoff TablesPayoff Tables

The consequence resulting from a The consequence resulting from a specific combination of a decision specific combination of a decision alternative and a state of nature is a alternative and a state of nature is a payoffpayoff..

A table showing payoffs for all A table showing payoffs for all combinations of decision alternatives combinations of decision alternatives and states of nature is a and states of nature is a payoff tablepayoff table..

Payoffs can be expressed in terms of Payoffs can be expressed in terms of profitprofit, , costcost, , timetime, , distancedistance or any other or any other appropriate measure.appropriate measure.

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Decision TreesDecision Trees

A A decision treedecision tree is a chronological is a chronological representation of the decision representation of the decision problem.problem.

Each decision tree has two types of Each decision tree has two types of nodes; nodes; round nodesround nodes correspond to correspond to the states of nature while the states of nature while square square nodesnodes correspond to the decision correspond to the decision alternatives. alternatives.

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The The branchesbranches leaving each round leaving each round node represent the different node represent the different states of nature while the states of nature while the branches leaving each square branches leaving each square node represent the different node represent the different decision alternatives.decision alternatives.

At the end of each limb of a tree At the end of each limb of a tree are the payoffs attained from the are the payoffs attained from the series of branches making up series of branches making up that limb. that limb.

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Decision Making without ProbabilitiesDecision Making without Probabilities

Three commonly used criteria for Three commonly used criteria for decision making when probability decision making when probability information regarding the information regarding the likelihood of the states of nature likelihood of the states of nature is unavailable are: is unavailable are:

•the the optimisticoptimistic approach approach

•the the conservativeconservative approach approach

•the the minimax regretminimax regret approach. approach.

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Optimistic ApproachOptimistic Approach

The The optimistic approachoptimistic approach would be would be used by an optimistic decision used by an optimistic decision maker.maker.

The The decision with the largest decision with the largest possible payoffpossible payoff is chosen. is chosen.

If the payoff table was in terms of If the payoff table was in terms of costs, the costs, the decision with the lowest decision with the lowest costcost would be chosen. would be chosen.

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Conservative ApproachConservative Approach

The The conservative approachconservative approach would be used by a would be used by a conservative decision maker. conservative decision maker.

For each decision the minimum payoff is listed For each decision the minimum payoff is listed and then the decision corresponding to the and then the decision corresponding to the maximum of these minimum payoffs is selected. maximum of these minimum payoffs is selected. (Hence, the (Hence, the minimum possible payoff is minimum possible payoff is maximizedmaximized.).)

If the payoff was in terms of costs, the maximum If the payoff was in terms of costs, the maximum costs would be determined for each decision and costs would be determined for each decision and then the decision corresponding to the minimum then the decision corresponding to the minimum of these maximum costs is selected. (Hence, of these maximum costs is selected. (Hence, the the maximum possible cost is minimizedmaximum possible cost is minimized.).)

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Minimax Regret ApproachMinimax Regret Approach

The minimax regret approach requires the The minimax regret approach requires the construction of a construction of a regret tableregret table or an or an opportunity loss tableopportunity loss table. .

This is done by calculating for each state of This is done by calculating for each state of nature the difference between each payoff nature the difference between each payoff and the largest payoff for that state of nature. and the largest payoff for that state of nature.

Then, using this regret table, the maximum Then, using this regret table, the maximum regret for each possible decision is listed. regret for each possible decision is listed.

The decision chosen is the one corresponding The decision chosen is the one corresponding to the to the minimum of the maximum regretsminimum of the maximum regrets..

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ExampleExample

Consider the following problem with three Consider the following problem with three decision alternatives and three states of nature decision alternatives and three states of nature with the following payoff table representing with the following payoff table representing profits:profits:

States of NatureStates of Nature

ss11 ss22 ss33

dd11 4 4 -2 4 4 -2

DecisionsDecisions dd22 0 3 -1 0 3 -1

dd33 1 5 - 1 5 -33

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Example: Optimistic ApproachExample: Optimistic Approach

An optimistic decision maker would use An optimistic decision maker would use the optimistic (maximax) approach. We the optimistic (maximax) approach. We choose the decision that has the largest single choose the decision that has the largest single value in the payoff table. value in the payoff table.

MaximumMaximum DecisionDecision PayoffPayoff

dd11 4 4

dd22 3 3

dd33 5 5

MaximaxMaximaxpayoffpayoff

MaximaxMaximaxdecisiondecision

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Formula SpreadsheetFormula SpreadsheetA B C D E F

123 Decision Maximum Recommended4 Alternative s1 s2 s3 Payoff Decision5 d1 4 4 -2 =MAX(B5:D5) =IF(E5=$E$9,A5,"")6 d2 0 3 -1 =MAX(B6:D6) =IF(E6=$E$9,A6,"")7 d3 1 5 -3 =MAX(B7:D7) =IF(E7=$E$9,A7,"")89 =MAX(E5:E7)

State of Nature

Best Payoff

PAYOFF TABLE

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Solution SpreadsheetSolution SpreadsheetA B C D E F

123 Decision Maximum Recommended4 Alternative s1 s2 s3 Payoff Decision5 d1 4 4 -2 46 d2 0 3 -1 37 d3 1 5 -3 5 d389 5

State of Nature

Best Payoff

PAYOFF TABLEA B C D E F

123 Decision Maximum Recommended4 Alternative s1 s2 s3 Payoff Decision5 d1 4 4 -2 46 d2 0 3 -1 37 d3 1 5 -3 5 d389 5

State of Nature

Best Payoff

PAYOFF TABLE

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Example: Conservative ApproachExample: Conservative Approach

A conservative decision maker would A conservative decision maker would use the conservative (maximin) approach. List use the conservative (maximin) approach. List the minimum payoff for each decision. the minimum payoff for each decision. Choose the decision with the maximum of Choose the decision with the maximum of these minimum payoffs.these minimum payoffs.

MinimumMinimum

DecisionDecision PayoffPayoff

dd11 -2 -2

dd22 -1 -1

dd33 -3 -3

MaximinMaximindecisiondecision

MaximinMaximinpayoffpayoff

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Formula SpreadsheetFormula Spreadsheet

A B C D E F123 Decision Minimum Recommended4 Alternative s1 s2 s3 Payoff Decision5 d1 4 4 -2 =MIN(B5:D5) =IF(E5=$E$9,A5,"")6 d2 0 3 -1 =MIN(B6:D6) =IF(E6=$E$9,A6,"")7 d3 1 5 -3 =MIN(B7:D7) =IF(E7=$E$9,A7,"")89 =MAX(E5:E7)

State of Nature

Best Payoff

PAYOFF TABLE

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Solution SpreadsheetSolution Spreadsheet

A B C D E F123 Decision Minimum Recommended4 Alternative s1 s2 s3 Payoff Decision5 d1 4 4 -2 -26 d2 0 3 -1 -1 d27 d3 1 5 -3 -389 -1

State of Nature

Best Payoff


123 Decision Minimum Recommended4 Alternative s1 s2 s3 Payoff Decision5 d1 4 4 -2 -26 d2 0 3 -1 -1 d27 d3 1 5 -3 -389 -1

State of Nature

Best Payoff

PAYOFF TABLE

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For the minimax regret approach, first For the minimax regret approach, first compute a regret table by subtracting each compute a regret table by subtracting each payoff in a column from the largest payoff in payoff in a column from the largest payoff in that column. In this example, in the first that column. In this example, in the first column subtract 4, 0, and 1 from 4; etc. The column subtract 4, 0, and 1 from 4; etc. The resulting regret table is: resulting regret table is:

ss11 ss22 ss33

dd11 0 1 1 0 1 1

dd22 4 2 0 4 2 0

dd33 3 0 2 3 0 2

Example: Minimax Regret ApproachExample: Minimax Regret Approach

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For each decision list the maximum For each decision list the maximum regret. Choose the decision with the minimum regret. Choose the decision with the minimum of these values.of these values.

MaximumMaximum DecisionDecision RegretRegret

dd11 1 1

dd22 4 4

dd33 3 3

Example: Minimax Regret ApproachExample: Minimax Regret Approach

MinimaxMinimaxdecisiondecision

MinimaxMinimaxregretregret

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Example: Minimax Regret Approach Example: Minimax Regret Approach


12 Decision 3 Altern. s1 s2 s34 d1 4 4 -25 d2 0 3 -16 d3 1 5 -3789 Decision Maximum Recommended10 Altern. s1 s2 s3 Regret Decision11 d1 =MAX($B$4:$B$6)-B4 =MAX($C$4:$C$6)-C4 =MAX($D$4:$D$6)-D4 =MAX(B11:D11) =IF(E11=$E$14,A11,"")12 d2 =MAX($B$4:$B$6)-B5 =MAX($C$4:$C$6)-C5 =MAX($D$4:$D$6)-D5 =MAX(B12:D12) =IF(E12=$E$14,A12,"")13 d3 =MAX($B$4:$B$6)-B6 =MAX($C$4:$C$6)-C6 =MAX($D$4:$D$6)-D6 =MAX(B13:D13) =IF(E13=$E$14,A13,"")14 =MIN(E11:E13)Minimax Regret Value

State of NaturePAYOFF TABLE

State of NatureOPPORTUNITY LOSS TABLE

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12 Decision 3 Alternative s1 s2 s34 d1 4 4 -25 d2 0 3 -16 d3 1 5 -3789 Decision Maximum Recommended10 Alternative s1 s2 s3 Regret Decision11 d1 0 1 1 1 d112 d2 4 2 0 413 d3 3 0 2 314 1Minimax Regret Value



A B C D E F12 Decision 3 Alternative s1 s2 s34 d1 4 4 -25 d2 0 3 -16 d3 1 5 -3789 Decision Maximum Recommended10 Alternative s1 s2 s3 Regret Decision11 d1 0 1 1 1 d112 d2 4 2 0 413 d3 3 0 2 314 1Minimax Regret Value



Example: Minimax Regret Approach Example: Minimax Regret Approach

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Decision Making with ProbabilitiesDecision Making with Probabilities

Expected Value ApproachExpected Value Approach

• If probabilistic information regarding the If probabilistic information regarding the states of nature is available, one may use states of nature is available, one may use the the expected value (EV) approachexpected value (EV) approach. .

• Here the expected return for each Here the expected return for each decision is calculated by summing the decision is calculated by summing the products of the payoff under each state of products of the payoff under each state of nature and the probability of the nature and the probability of the respective state of nature occurring. respective state of nature occurring.

• The decision yielding the The decision yielding the best expected best expected returnreturn is chosen. is chosen.

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The The expected value of a decision alternativeexpected value of a decision alternative is the is the sum of weighted payoffs for the decision sum of weighted payoffs for the decision alternative.alternative.

The expected value (EV) of decision alternative The expected value (EV) of decision alternative ddii is defined as:is defined as:

where: where: NN = the number of states of nature = the number of states of nature

PP((ssj j ) = the probability of state of nature ) = the probability of state of nature ssjj

VVij ij = the payoff corresponding to = the payoff corresponding to decision decision alternative alternative ddii and state of and state of nature nature ssjj

Expected Value of a Decision AlternativeExpected Value of a Decision Alternative

EV( ) ( )d P s Vi j ijj

N

1

EV( ) ( )d P s Vi j ijj

N

1

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Example: Burger PrinceExample: Burger Prince

Burger Prince Restaurant is considering Burger Prince Restaurant is considering opening a new restaurant on Main Street. It opening a new restaurant on Main Street. It has threehas three

different models, each with a differentdifferent models, each with a different

seating capacity. Burger Princeseating capacity. Burger Prince

estimates that the average number ofestimates that the average number of

customers per hour will be 80, 100, orcustomers per hour will be 80, 100, or

120. The payoff table for the three120. The payoff table for the three

models is on the next slide. models is on the next slide.

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Payoff TablePayoff Table

Average Number of Customers Per Average Number of Customers Per HourHour

ss11 = 80 = 80 ss22 = 100 = 100 ss33 = 120 = 120

Model A $10,000 $15,000 Model A $10,000 $15,000 $14,000$14,000

Model B $ 8,000 $18,000 Model B $ 8,000 $18,000 $12,000$12,000

Model C $ 6,000 $16,000 Model C $ 6,000 $16,000 $21,000$21,000

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Calculate the expected value for each Calculate the expected value for each decision. The decision tree on the next slide decision. The decision tree on the next slide can assist in this calculation. Here can assist in this calculation. Here dd11, , dd22, , dd3 3

represent the decision alternatives of models represent the decision alternatives of models A, B, C, and A, B, C, and ss11, , ss22, , ss3 3 represent the states of represent the states of

nature of 80, 100, and 120.nature of 80, 100, and 120.

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Decision TreeDecision Tree

1111

.2.2

.4.4

.4.4

.4.4

.2.2

.4.4

.4.4

.2.2

.4.4

dd11

dd22

dd33

ss11

ss11

ss11

ss22

ss33

ss22

ss22

ss33

ss33

PayoffsPayoffs

10,00010,000

15,00015,000

14,00014,0008,0008,000

18,00018,000

12,00012,000

6,0006,000

16,00016,000

21,00021,000

2222

3333

4444

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Expected Value for Each DecisionExpected Value for Each Decision

Choose the model with largest EV, Model Choose the model with largest EV, Model C.C.

3333

dd11

dd22

dd33

EMV = .4(10,000) + .2(15,000) + .4(14,000)EMV = .4(10,000) + .2(15,000) + .4(14,000) = $12,600= $12,600

EMV = .4(8,000) + .2(18,000) + .4(12,000)EMV = .4(8,000) + .2(18,000) + .4(12,000) = $11,600= $11,600

EMV = .4(6,000) + .2(16,000) + .4(21,000)EMV = .4(6,000) + .2(16,000) + .4(21,000) = $14,000= $14,000

Model AModel A

Model BModel B

Model CModel C

2222

1111

4444

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123 Decision Expected Recommended4 Alternative s1 = 80 s2 = 100 s3 = 120 Value Decision5 d1 = Model A 10,000 15,000 14,000 =$B$8*B5+$C$8*C5+$D$8*D5 =IF(E5=$E$9,A5,"")6 d2 = Model B 8,000 18,000 12,000 =$B$8*B6+$C$8*C6+$D$8*D6 =IF(E6=$E$9,A6,"")7 d3 = Model C 6,000 16,000 21,000 =$B$8*B7+$C$8*C7+$D$8*D7 =IF(E7=$E$9,A7,"")8 Probability 0.4 0.2 0.49 =MAX(E5:E7)

State of Nature

Maximum Expected Value


123 Decision Expected Recommended4 Alternative s1 = 80 s2 = 100 s3 = 120 Value Decision5 d1 = Model A 10,000 15,000 14,000 =$B$8*B5+$C$8*C5+$D$8*D5 =IF(E5=$E$9,A5,"")6 d2 = Model B 8,000 18,000 12,000 =$B$8*B6+$C$8*C6+$D$8*D6 =IF(E6=$E$9,A6,"")7 d3 = Model C 6,000 16,000 21,000 =$B$8*B7+$C$8*C7+$D$8*D7 =IF(E7=$E$9,A7,"")8 Probability 0.4 0.2 0.49 =MAX(E5:E7)

State of Nature


PAYOFF TABLE

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123 Decision Expected Recommended4 Alternative s1 = 80 s2 = 100 s3 = 120 Value Decision5 d1 = Model A 10,000 15,000 14,000 126006 d2 = Model B 8,000 18,000 12,000 116007 d3 = Model C 6,000 16,000 21,000 14000 d3 = Model C8 Probability 0.4 0.2 0.49 14000

State of Nature



123 Decision Expected Recommended4 Alternative s1 = 80 s2 = 100 s3 = 120 Value Decision5 d1 = Model A 10,000 15,000 14,000 126006 d2 = Model B 8,000 18,000 12,000 116007 d3 = Model C 6,000 16,000 21,000 14000 d3 = Model C8 Probability 0.4 0.2 0.49 14000

State of Nature


PAYOFF TABLE


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Expected Value of Perfect InformationExpected Value of Perfect Information

Frequently information is available which Frequently information is available which can improve the probability estimates for can improve the probability estimates for the states of nature. the states of nature.

The The expected value of perfect informationexpected value of perfect information (EVPI) is the increase in the expected profit (EVPI) is the increase in the expected profit that would result if one knew with certainty that would result if one knew with certainty which state of nature would occur. which state of nature would occur.

The EVPI provides an The EVPI provides an upper bound on the upper bound on the expected value of any sample or survey expected value of any sample or survey informationinformation. .

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EVPI CalculationEVPI Calculation

• Step 1:Step 1:

Determine the optimal return Determine the optimal return corresponding to each state of nature.corresponding to each state of nature.

• Step 2:Step 2:

Compute the expected value of these Compute the expected value of these optimal returns.optimal returns.

• Step 3:Step 3:

Subtract the EV of the optimal Subtract the EV of the optimal decision from the amount determined in step decision from the amount determined in step (2).(2).

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Calculate the expected value for the Calculate the expected value for the optimum payoff for each state of nature and optimum payoff for each state of nature and subtract the EV of the optimal decision.subtract the EV of the optimal decision.

EVPI= .4(10,000) + .2(18,000) + .4(21,000) - EVPI= .4(10,000) + .2(18,000) + .4(21,000) - 14,000 = $2,00014,000 = $2,000


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SpreadsheetSpreadsheetA B C D E F

123 Decision Expected Recommended4 Alternative s1 = 80 s2 = 100 s3 = 120 Value Decision5 d1 = Model A 10,000 15,000 14,000 126006 d2 = Model B 8,000 18,000 12,000 116007 d3 = Model C 6,000 16,000 21,000 14000 d3 = Model C8 Probability 0.4 0.2 0.49 140001011 EVwPI EVPI12 10,000 18,000 21,000 16000 2000

State of Nature


PAYOFF TABLE

Maximum Payoff

A B C D E F123 Decision Expected Recommended4 Alternative s1 = 80 s2 = 100 s3 = 120 Value Decision5 d1 = Model A 10,000 15,000 14,000 126006 d2 = Model B 8,000 18,000 12,000 116007 d3 = Model C 6,000 16,000 21,000 14000 d3 = Model C8 Probability 0.4 0.2 0.49 140001011 EVwPI EVPI12 10,000 18,000 21,000 16000 2000

State of Nature


PAYOFF TABLE

Maximum Payoff


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Risk AnalysisRisk Analysis

Risk analysisRisk analysis helps the decision maker helps the decision maker recognize the difference between:recognize the difference between:

• the expected value of a decision alternative, the expected value of a decision alternative, andand

• the payoff that might actually occurthe payoff that might actually occur The The risk profilerisk profile for a decision alternative shows for a decision alternative shows

the possible payoffs for the decision the possible payoffs for the decision alternative along with their associated alternative along with their associated probabilities.probabilities.

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Risk ProfileRisk Profile

Model C Decision AlternativeModel C Decision Alternative

.10.10

.20.20

.30.30

.40.40

.50.50

5 10 15 20 255 10 15 20 25

Pro

bab

ility

Pro

bab

ility

Profit ($thousands)Profit ($thousands)

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Sensitivity AnalysisSensitivity Analysis

Sensitivity analysisSensitivity analysis can be used to determine can be used to determine how changes to the following inputs affect the how changes to the following inputs affect the recommended decision alternative:recommended decision alternative:

• probabilities for the states of natureprobabilities for the states of nature

• values of the payoffsvalues of the payoffs If a small change in the value of one of the If a small change in the value of one of the

inputs causes a change in the recommended inputs causes a change in the recommended decision alternative, extra effort and care decision alternative, extra effort and care should be taken in estimating the input value.should be taken in estimating the input value.

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Bayes’ Theorem and Posterior Bayes’ Theorem and Posterior ProbabilitiesProbabilities

Knowledge of sample (survey) information can be Knowledge of sample (survey) information can be used to revise the probability estimates for the used to revise the probability estimates for the states of nature. states of nature.

Prior to obtaining this information, the probability Prior to obtaining this information, the probability estimates for the states of nature are called estimates for the states of nature are called prior prior probabilitiesprobabilities. .

With knowledge of With knowledge of conditional probabilitiesconditional probabilities for the for the outcomes or indicators of the sample or survey outcomes or indicators of the sample or survey information, these prior probabilities can be information, these prior probabilities can be revised by employing revised by employing Bayes' TheoremBayes' Theorem. .

The outcomes of this analysis are called The outcomes of this analysis are called posterior posterior probabilitiesprobabilities or or branch probabilitiesbranch probabilities for decision for decision trees.trees.

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Computing Branch ProbabilitiesComputing Branch Probabilities

Branch (Posterior) Probabilities CalculationBranch (Posterior) Probabilities Calculation

• Step 1:Step 1:

For each state of nature, multiply the For each state of nature, multiply the prior probability by its conditional probability prior probability by its conditional probability for the indicator -- this gives the for the indicator -- this gives the joint joint probabilitiesprobabilities for the states and indicator. for the states and indicator.

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Computing Branch ProbabilitiesComputing Branch Probabilities

Branch (Posterior) Probabilities CalculationBranch (Posterior) Probabilities Calculation

• Step 2:Step 2:

Sum these joint probabilities over all Sum these joint probabilities over all states -- this gives the states -- this gives the marginal probabilitymarginal probability for for the indicator.the indicator.

• Step 3:Step 3:

For each state, divide its joint For each state, divide its joint probability by the marginal probability for the probability by the marginal probability for the indicator -- this gives the posterior probability indicator -- this gives the posterior probability distribution.distribution.

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Expected Value of Sample InformationExpected Value of Sample Information

The The expected value of sample informationexpected value of sample information (EVSI) is the additional expected profit (EVSI) is the additional expected profit possible through knowledge of the sample or possible through knowledge of the sample or survey information. survey information.

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EVSI CalculationEVSI Calculation

• Step 1:Step 1:

Determine the optimal decision and its Determine the optimal decision and its expected return for the possible outcomes of expected return for the possible outcomes of the sample using the posterior probabilities for the sample using the posterior probabilities for the states of nature. the states of nature.

• Step 2:Step 2:

Compute the expected value of these Compute the expected value of these optimal returns.optimal returns.

• Step 3:Step 3:

Subtract the EV of the optimal decision Subtract the EV of the optimal decision obtained without using the sample information obtained without using the sample information from the amount determined in step (2).from the amount determined in step (2).

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Efficiency of Sample InformationEfficiency of Sample Information

Efficiency of sample informationEfficiency of sample information is the ratio of is the ratio of EVSI to EVPI. EVSI to EVPI.

As the EVPI provides an upper bound for the As the EVPI provides an upper bound for the EVSI, efficiency is always a number between 0 EVSI, efficiency is always a number between 0 and 1.and 1.

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Burger Prince must decide whether or not Burger Prince must decide whether or not to purchase a marketing survey from Stanton to purchase a marketing survey from Stanton Marketing for $1,000. The results of the survey Marketing for $1,000. The results of the survey are "favorable" or "unfavorable". The conditional are "favorable" or "unfavorable". The conditional probabilities are:probabilities are:

P(favorable | 80 customers per hour) P(favorable | 80 customers per hour) = .2= .2

P(favorable | 100 customers per hour) P(favorable | 100 customers per hour) = .5 = .5

P(favorable | 120 customers per hour) P(favorable | 120 customers per hour) = .9 = .9

Should Burger Prince have the survey Should Burger Prince have the survey performed by Stanton Marketing?performed by Stanton Marketing?

Sample InformationSample Information

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Influence DiagramInfluence Diagram

RestaurantRestaurantSizeSize ProfitProfit

Avg. NumberAvg. Numberof Customersof Customers

Per HourPer Hour

MarketMarketSurveySurveyResultsResults

MarketMarketSurveySurvey

DecisionDecisionChanceChanceConsequenceConsequence

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FavorableFavorable

StateState PriorPrior ConditionalConditional JointJoint PosteriorPosterior

80 .4 .2 .08 .14880 .4 .2 .08 .148

100 .2 .5 .10 .185100 .2 .5 .10 .185

120 .4 .9 120 .4 .9 .36.36 .667.667

Total .54 1.000Total .54 1.000

P(favorable) = .54P(favorable) = .54

Posterior ProbabilitiesPosterior Probabilities

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UnfavorableUnfavorable

StateState PriorPrior ConditionalConditional JointJoint PosteriorPosterior

80 .4 .8 .32 .69680 .4 .8 .32 .696

100 .2 .5 .10 .217100 .2 .5 .10 .217

120 .4 .1 120 .4 .1 .04.04 .087.087

Total .46 1.000Total .46 1.000

P(unfavorable) = .46P(unfavorable) = .46


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Formula SpreadsheetFormula SpreadsheetA B C D E

12 Prior Conditional Joint Posterior3 State of Nature Probabilities Probabilities Probabilities Probabilities4 s1 = 80 0.4 0.2 =B4*C4 =D4/$D$75 s2 = 100 0.2 0.5 =B5*C5 =D5/$D$76 s3 = 120 0.4 0.9 =B6*C6 =D6/$D$77 =SUM(D4:D6)8910 Prior Conditional Joint Posterior11 State of Nature Probabilities Probabilities Probabilities Probabilities12 s1 = 80 0.4 0.8 =B12*C12 =D12/$D$1513 s2 = 100 0.2 0.5 =B13*C13 =D13/$D$1514 s3 = 120 0.4 0.1 =B14*C14 =D14/$D$1515 =SUM(D12:D14)

Market Research Favorable

P(Favorable) =

Market Research Unfavorable

P(Unfavorable) =


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Solution SpreadsheetSolution SpreadsheetA B C D E

12 Prior Conditional Joint Posterior3 State of Nature Probabilities Probabilities Probabilities Probabilities4 s1 = 80 0.4 0.2 0.08 0.1485 s2 = 100 0.2 0.5 0.10 0.1856 s3 = 120 0.4 0.9 0.36 0.6677 0.548910 Prior Conditional Joint Posterior11 State of Nature Probabilities Probabilities Probabilities Probabilities12 s1 = 80 0.4 0.8 0.32 0.69613 s2 = 100 0.2 0.5 0.10 0.21714 s3 = 120 0.4 0.1 0.04 0.08715 0.46

Market Research Favorable

P(Favorable) =

Market Research Unfavorable

P(Favorable) =


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Top HalfTop Half

ss11 (.148) (.148)

ss1 1 (.148)(.148)

ss11 (.148) (.148)

ss22 (.185) (.185)

ss22 (.185) (.185)

ss22 (.185) (.185)

ss33 (.667) (.667)

ss3 3 (.667)(.667)

ss33 (.667) (.667)

$10,000$10,000

$15,000$15,000

$14,000$14,000$8,000$8,000

$18,000$18,000

$12,000$12,000$6,000$6,000

$16,000$16,000

$21,000$21,000

II11

(.54)(.54)

dd11

dd22

dd33

2222

4444

5555

6666

1111

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Bottom HalfBottom Half

ss11 (.696) (.696)

ss11 (.696) (.696)

ss11 (.696) (.696)

ss22 (.217) (.217)

ss22 (.217) (.217)

ss22 (.217) (.217)

ss33 (.087) (.087)

ss3 3 (.087)(.087)

ss33 (.087) (.087)

$10,000$10,000

$15,000$15,000

$18,000$18,000

$14,000$14,000$8,000$8,000

$12,000$12,000$6,000$6,000

$16,000$16,000

$21,000$21,000

II22

(.46)(.46) dd11

dd22

dd33

7777

9999

88883333

1111


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II22

(.46)(.46)

dd11

dd22

dd33

EMV = .696(10,000) + .217(15,000)EMV = .696(10,000) + .217(15,000) +.087(14,000)= +.087(14,000)= $11,433$11,433

EMV = .696(8,000) + .217(18,000)EMV = .696(8,000) + .217(18,000) + .087(12,000) = $10,554+ .087(12,000) = $10,554

EMV = .696(6,000) + .217(16,000)EMV = .696(6,000) + .217(16,000) +.087(21,000) = $9,475+.087(21,000) = $9,475

II11

(.54)(.54)

dd11

dd22

dd33

EMV = .148(10,000) + .185(15,000)EMV = .148(10,000) + .185(15,000) + .667(14,000) = $13,593+ .667(14,000) = $13,593

EMV = .148 (8,000) + .185(18,000)EMV = .148 (8,000) + .185(18,000) + .667(12,000) = $12,518+ .667(12,000) = $12,518

EMV = .148(6,000) + .185(16,000)EMV = .148(6,000) + .185(16,000) +.667(21,000) = +.667(21,000) = $17,855$17,855

4444

5555

6666

7777

8888

9999

2222

3333

1111

$17,855$17,855

$11,433$11,433


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If the outcome of the survey is "favorable”,If the outcome of the survey is "favorable”,

choose Model C. If it is “unfavorable”, choose choose Model C. If it is “unfavorable”, choose Model A.Model A.

EVSI = .54($17,855) + .46($11,433) - $14,000 EVSI = .54($17,855) + .46($11,433) - $14,000 = $900.88 = $900.88

Since this is less than the cost of the Since this is less than the cost of the survey, the survey should survey, the survey should notnot be purchased. be purchased.


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Efficiency of Sample InformationEfficiency of Sample Information

The efficiency of the survey:The efficiency of the survey:

EVSI/EVPI = ($900.88)/($2000) = .4504EVSI/EVPI = ($900.88)/($2000) = .4504

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Bayes’ Decision Rule:

Using the best available estimates of the

probabilities of the respective states of nature

(currently the prior probabilities), calculate the

expected value of the payoff for each of the

possible actions. Choose the action with the

maximum expected payoff.

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Bayes’ theorySi: State of Nature (i = 1 ~ n)

P(Si): Prior Probability

Ij: Professional Information (Experiment)( j = 1 ~ n)

P(Ij | Si): Conditional Probability

P(Ij Si) = P(Si Ij): Joint Probability

P(Si | Ij): Posterior Probability

P(Si | Ij)

n

1iiij

iij

j

ji

)S(P)S|I(P

)S(P)S|I(P

)I(P

)IS(P

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Home WorkHome Work

14-2014-20

Due Day: Nov 7Due Day: Nov 7

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End of Chapter 14End of Chapter 14

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EMGT 501 HW Solutions Chapter 12 - SELF TEST 9 Chapter 12 - SELF TEST 18

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