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ChapterChapter 88Decision 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 InformationDecision Analysis with Sample Information Computing Branch ProbabilitiesComputing Branch Probabilities
Problem FormulationProblem Formulation
A decision problem is characterized by decisionA decision problem is characterized by decisionalternatives, states of nature, and resulting payoffs.alternatives, states of nature, and resulting payoffs.
TheThe decision alternativesdecision alternatives are the different possibleare the different possiblestrategies the decision maker can employ.strategies the decision maker can employ.
TheThe states of naturestates of nature refer to future events, notrefer to future events, notunder the control of the decision maker, whichunder the control of the decision maker, whichmay occur. States of nature should be defined somay occur. States of nature should be defined sothat they are mutually exclusive and collectivelythat they are mutually exclusive and collectivelyexhaustive.exhaustive.
Influence DiagramsInfluence Diagrams
AnAn influence diagraminfluence diagram is a graphical device showingis a graphical device showingthe relationships among the decisions, the chancethe relationships among the decisions, the chanceevents, and the consequences.events, and the consequences.
Squares or rectanglesSquares or rectangles depict decision nodes.depict decision 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 the directionconnecting the nodes show the direction
of influence.of influence.
22
Payoff TablesPayoff Tables
The consequence resulting from a specificThe consequence resulting from a specificcombination of a decision alternative and a state ofcombination of a decision alternative and a state ofnature is anature is a payoffpayoff..
A table showing payoffs for all combinations ofA table showing payoffs for all combinations ofdecision alternatives and states of nature is adecision alternatives and states of nature is a payoffpayofftabletable..
Payoffs can be expressed in terms ofPayoffs can be expressed in terms of profitprofit,, costcost,, timetime,,distancedistance or any other appropriate measure.or any other appropriate measure.
Decision TreesDecision Trees
AA decision treedecision tree is a chronological representation ofis a chronological representation ofthe decision problem.the decision problem.
Each decision tree has two types of nodes;Each decision tree has two types of nodes; roundroundnodesnodes correspond to the states of nature whilecorrespond to the states of nature while squaresquarenodesnodes correspond to the decision alternatives.correspond to the decision alternatives.
TheThe branchesbranches leaving each round node represent theleaving each round node represent thedifferent states of nature while the branches leavingdifferent states of nature while the branches leavingeach square node represent the different decisioneach square node represent the different decisionalternatives.alternatives.
At the end of each limb of a tree are the payoffsAt the end of each limb of a tree are the payoffsattained from the series of branches making up thatattained from the series of branches making up thatlimb.limb.
Decision Making without ProbabilitiesDecision Making without Probabilities
Three commonly used criteria for decision makingThree commonly used criteria for decision makingwhen probability information regarding thewhen probability information regarding thelikelihood of the states of nature is unavailable are:likelihood of the states of nature is unavailable are:•• thethe optimisticoptimistic approachapproach•• thethe conservativeconservative approachapproach•• thethe minimax regretminimax regret approach.approach.
33
Optimistic ApproachOptimistic Approach
TheThe optimistic approachoptimistic approach would be used by anwould be used by anoptimistic decision maker.optimistic decision maker.
TheThe decision with the largest possible payoffdecision with the largest possible payoff isischosen.chosen.
If the payoff table was in terms of costs, theIf the payoff table was in terms of costs, the decisiondecisionwith the lowest costwith the lowest cost would be chosen.would be chosen.
Conservative ApproachConservative Approach
TheThe conservative approachconservative approach would be used by awould be used by aconservative decision maker.conservative decision maker.
For each decision the minimum payoff is listed andFor each decision the minimum payoff is listed andthen the decision corresponding to the maximumthen the decision corresponding to the maximumof these minimum payoffs is selected. (Hence, theof these minimum payoffs is selected. (Hence, theminimum possible payoff is maximizedminimum possible payoff is maximized.).)
If the payoff was in terms of costs, the maximumIf the payoff was in terms of costs, the maximumcosts would be determined for each decision andcosts would be determined for each decision andthen the decision corresponding to the minimumthen the decision corresponding to the minimumof these maximum costs is selected. (Hence, theof these maximum costs is selected. (Hence, themaximum possible cost is minimizedmaximum possible cost is minimized.).)
Minimax Regret ApproachMinimax Regret Approach
The minimax regret approach requires theThe minimax regret approach requires theconstruction of aconstruction of a regret tableregret table or anor an opportunityopportunityloss tableloss table..
This is done by calculating for each state of natureThis is done by calculating for each state of naturethe difference between each payoff and the largestthe difference between each payoff and the largestpayoff for that state of nature.payoff for that state of nature.
Then, using this regret table, the maximum regretThen, using this regret table, the maximum regretfor each possible decision is listed.for each possible decision is listed.
The decision chosen is the one corresponding toThe decision chosen is the one corresponding tothethe minimum of the maximum regretsminimum of the maximum regrets..
44
ExampleExample
Consider the following problem with threeConsider the following problem with threedecision alternatives and three states of nature withdecision alternatives and three states of nature withthe following payoff table representing profits:the following payoff table representing profits:
States of NatureStates of Naturess11 ss22 ss33
dd11 4 44 4 --22DecisionsDecisions dd22 0 30 3 --11
dd33 1 51 5 --33
Example: Optimistic ApproachExample: Optimistic Approach
An optimistic decision maker would use theAn optimistic decision maker would use theoptimistic (maximax) approach. We choose theoptimistic (maximax) approach. We choose thedecision that has the largest single value in thedecision that has the largest single value in thepayoff table.payoff table.
MaximumMaximumDecisionDecision PayoffPayoffdd11 44dd22 33dd33 55
MaximaxMaximaxpayoffpayoff
MaximaxMaximaxdecisiondecision
Example: Optimistic ApproachExample: Optimistic Approach
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$9,A7,"")89 =MAX(E5:E7)
State of Nature
Best Payoff
PAYOFF TABLE
55
Example: Optimistic ApproachExample: Optimistic Approach
Solution SpreadsheetSolution SpreadsheetA B C D E F
123 Decision Maximum Recommended4 Alternative s1 s2 s3 Payoff Decision5 d1 4 4 -2 46 d2 0 3 -1 37 d3 1 5 -3 5 d389 5
State of Nature
Best Payoff
PAYOFF TABLE
Example: Conservative ApproachExample: Conservative Approach
A conservative decision maker would use theA conservative decision maker would use theconservative (maximin) approach. List the minimumconservative (maximin) approach. List the minimumpayoff for each decision. Choose the decision withpayoff for each decision. Choose the decision withthe maximum of these minimum payoffs.the maximum of these minimum payoffs.
MinimumMinimumDecisionDecision PayoffPayoffdd11 --22dd22 --11dd33 --33
MaximinMaximindecisiondecision
MaximinMaximinpayoffpayoff
Example: Conservative ApproachExample: Conservative Approach
Formula SpreadsheetFormula SpreadsheetA B C D E F
123 Decision Minimum Recommended4 Alternative s1 s2 s3 Payoff Decision5 d1 4 4 -2 =MIN(B5:D5) =IF(E5=$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$9,A7,"")89 =MAX(E5:E7)
State of Nature
Best Payoff
PAYOFF TABLE
66
Example: Conservative ApproachExample: Conservative Approach
Solution SpreadsheetSolution SpreadsheetA B C D E F
123 Decision Minimum Recommended4 Alternative s1 s2 s3 Payoff Decision5 d1 4 4 -2 -26 d2 0 3 -1 -1 d27 d3 1 5 -3 -389 -1
State of Nature
Best Payoff
PAYOFF TABLE
For the minimax regret approach, first compute aFor the minimax regret approach, first compute aregret table by subtracting each payoff in a columnregret table by subtracting each payoff in a columnfrom the largest payoff in that column. In thisfrom the largest payoff in that column. In thisexample, in the first column subtract 4, 0, and 1 fromexample, in the first column subtract 4, 0, and 1 from4; etc. The resulting regret table is:4; etc. The resulting regret table is:
ss11 ss22 ss33
dd11 0 1 10 1 1dd22 4 2 04 2 0dd33 3 0 23 0 2
Example: Minimax Regret ApproachExample: Minimax Regret Approach
For each decision list the maximum regret.For each decision list the maximum regret.Choose the decision with the minimum of theseChoose the decision with the minimum of thesevalues.values.
MaximumMaximumDecisionDecision RegretRegretdd11 11dd22 44dd33 33
Example: Minimax Regret ApproachExample: Minimax Regret Approach
MinimaxMinimaxdecisiondecision
MinimaxMinimaxregretregret
77
Example: Minimax Regret ApproachExample: Minimax Regret Approach
Formula SpreadsheetFormula SpreadsheetA B C D E F
12 Decision3 Altern. s1 s2 s34 d1 4 4 -25 d2 0 3 -16 d3 1 5 -3789 Decision Maximum Recommended10 Altern. s1 s2 s3 Regret Decision11 d1 =MAX($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
Solution SpreadsheetSolution SpreadsheetA B C D E F
12 Decision3 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
State of NaturePAYOFF TABLE
State of NatureOPPORTUNITY LOSS TABLE
Example: Minimax Regret ApproachExample: Minimax Regret Approach
Decision Making with ProbabilitiesDecision Making with Probabilities
Expected Value ApproachExpected Value Approach•• If probabilistic information regarding the statesIf probabilistic information regarding the states
of nature is available, one may use theof nature is available, one may use the expectedexpectedvalue (EV) approachvalue (EV) approach..
•• Here the expected return for each decision isHere the expected return for each decision iscalculated by summing the products of thecalculated by summing the products of thepayoff under each state of nature and thepayoff under each state of nature and theprobability of the respective state of natureprobability of the respective state of natureoccurring.occurring.
•• The decision yielding theThe decision yielding the best expected returnbest expected return isischosen.chosen.
88
TheThe expected value of a decision alternativeexpected value of a decision alternative is theis thesum of weighted payoffs for the decision alternative.sum of weighted payoffs for the decision alternative.
The expected value (EV) of decision alternativeThe expected value (EV) of decision alternative ddii isisdefined as:defined as:
where:where: NN = the number of states of nature= the number of states of naturePP((ssjj ) = the probability of state of nature) = the probability of state of nature ssjjVVijij = the payoff corresponding to decision= the payoff corresponding to decision
alternativealternative ddii and state of natureand state of 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
Example: Burger PrinceExample: Burger Prince
Burger Prince Restaurant is considering openingBurger Prince Restaurant is considering openinga new restaurant on Main Street. It has threea new restaurant on Main Street. It has threedifferent models, each with a differentdifferent models, each with a differentseating capacity. Burger Princeseating capacity. Burger Princeestimates that the average number ofestimates that the average number ofcustomers per hour will be 80, 100, orcustomers per hour will be 80, 100, or120. The payoff table for the three120. The payoff table for the threemodels is on the next slide.models is on the next slide.
Payoff TablePayoff Table
Average Number of Customers Per HourAverage Number of Customers Per Hourss11 = 80= 80 ss22 = 100= 100 ss33 = 120= 120
Model A $10,000 $15,000 $14,000Model A $10,000 $15,000 $14,000Model B $ 8,000 $18,000 $12,000Model B $ 8,000 $18,000 $12,000Model C $ 6,000 $16,000 $21,000Model C $ 6,000 $16,000 $21,000
99
Expected Value ApproachExpected Value Approach
Calculate the expected value for each decision.Calculate the expected value for each decision.The decision tree on the next slide can assist in thisThe decision tree on the next slide can assist in thiscalculation. Herecalculation. Here dd11,, dd22,, dd33 represent the decisionrepresent the decisionalternatives of models A, B, C, andalternatives of models A, B, C, and ss11,, ss22,, ss33 representrepresentthe states of nature of 80, 100, and 120.the states of nature of 80, 100, and 120.
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
ss22ss33
ss22
ss22ss33
ss33
PayoffsPayoffs10,00010,00015,00015,00014,00014,0008,0008,000
18,00018,00012,00012,0006,0006,000
16,00016,00021,00021,000
2222
3333
4444
Expected Value for Each DecisionExpected Value for Each Decision
Choose the model with largest EV, Model C.Choose the model with largest EV, Model 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= $14,000
Model AModel A
Model BModel B
Model CModel C
2222
1111
4444
1010
Expected Value ApproachExpected Value Approach
Formula SpreadsheetFormula SpreadsheetA B C D E F
123 Decision Expected Recommended4 Alternative s1 = 80 s2 = 100 s3 = 120 Value Decision5 d1 = Model A 10,000 15,000 14,000 =$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
PAYOFF TABLE
Solution SpreadsheetSolution SpreadsheetA 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 14000
State of Nature
Maximum Expected Value
PAYOFF TABLE
Expected Value ApproachExpected Value Approach
Expected Value of Perfect InformationExpected Value of Perfect Information
Frequently information is available which canFrequently information is available which canimprove the probability estimates for the states ofimprove the probability estimates for the states ofnature.nature.
TheThe expected value of perfect informationexpected value of perfect information (EVPI) is(EVPI) isthe increase in the expected profit that wouldthe increase in the expected profit that wouldresult if one knew with certainty which state ofresult if one knew with certainty which state ofnature would occur.nature would occur.
The EVPI provides anThe EVPI provides an upper bound on theupper bound on theexpected value of any sample or surveyexpected value of any sample or surveyinformationinformation..
1111
Expected Value of Perfect InformationExpected Value of Perfect Information
EVPI CalculationEVPI Calculation•• Step 1:Step 1:
Determine the optimal return corresponding toDetermine the optimal return corresponding toeach state of nature.each state of nature.
•• Step 2:Step 2:Compute the expected value of these optimalCompute the expected value of these optimal
returns.returns.•• Step 3:Step 3:
Subtract the EV of the optimal decision from theSubtract the EV of the optimal decision from theamount determined in step (2).amount determined in step (2).
Calculate the expected value for the optimumCalculate the expected value for the optimumpayoff for each state of nature and subtract the EV ofpayoff for each state of nature and subtract the EV ofthe optimal decision.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
Expected Value of Perfect InformationExpected Value of Perfect Information
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
Maximum Expected Value
PAYOFF TABLE
Maximum Payoff
Expected Value of Perfect InformationExpected Value of Perfect Information
1212
Risk AnalysisRisk Analysis
Risk analysisRisk analysis helps the decision maker recognize thehelps the decision maker recognize thedifference between:difference between:•• the expected value of a decision alternative, andthe expected value of a decision alternative, and•• the payoff that might actually occurthe payoff that might actually occur
TheThe risk profilerisk profile for a decision alternative shows thefor a decision alternative shows thepossible payoffs for the decision alternative alongpossible payoffs for the decision alternative alongwith their associated probabilities.with their associated probabilities.
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
Prob
abili
tyPr
obab
ility
Sensitivity AnalysisSensitivity Analysis
Sensitivity analysisSensitivity analysis can be used to determine howcan be used to determine howchanges to the following inputs affect thechanges to the following inputs affect therecommended 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 inputsIf a small change in the value of one of the inputscauses a change in the recommended decisioncauses a change in the recommended decisionalternative, extra effort and care should be taken inalternative, extra effort and care should be taken inestimating the input value.estimating the input value.
1313
Bayes’ Theorem and Posterior ProbabilitiesBayes’ Theorem and Posterior Probabilities
Knowledge of sample (survey) information can be usedKnowledge of sample (survey) information can be usedto revise the probability estimates for the states of nature.to revise the probability estimates for the states of nature.
Prior to obtaining this information, the probabilityPrior to obtaining this information, the probabilityestimates for the states of nature are calledestimates for the states of nature are called priorpriorprobabilitiesprobabilities..
With knowledge ofWith knowledge of conditional probabilitiesconditional probabilities for thefor theoutcomes or indicators of the sample or surveyoutcomes or indicators of the sample or surveyinformation, these prior probabilities can be revised byinformation, these prior probabilities can be revised byemployingemploying Bayes' TheoremBayes' Theorem..
The outcomes of this analysis are calledThe outcomes of this analysis are called posteriorposteriorprobabilitiesprobabilities oror branch probabilitiesbranch probabilities for decision trees.for decision trees.
Computing Branch ProbabilitiesComputing Branch Probabilities
Branch (Posterior) Probabilities CalculationBranch (Posterior) Probabilities Calculation•• Step 1:Step 1:
For each state of nature, multiply the priorFor each state of nature, multiply the priorprobability by its conditional probability for theprobability by its conditional probability for theindicatorindicator ---- this gives thethis gives the joint probabilitiesjoint probabilities for thefor thestates and indicator.states and indicator.
Computing Branch ProbabilitiesComputing Branch Probabilities
Branch (Posterior) Probabilities CalculationBranch (Posterior) Probabilities Calculation•• Step 2:Step 2:
Sum these joint probabilities over all statesSum these joint probabilities over all states ---- thisthisgives thegives the marginal probabilitymarginal probability for the indicator.for the indicator.
•• Step 3:Step 3:For each state, divide its joint probability by theFor each state, divide its joint probability by the
marginal probability for the indicatormarginal probability for the indicator ---- this givesthis givesthe posterior probability distribution.the posterior probability distribution.
1414
Expected Value of Sample InformationExpected Value of Sample Information
TheThe expected value of sample informationexpected value of sample information (EVSI) is(EVSI) isthe additional expected profit possible throughthe additional expected profit possible throughknowledge of the sample or survey information.knowledge of the sample or survey information.
Expected Value of Sample InformationExpected Value of Sample Information
EVSI CalculationEVSI Calculation•• Step 1:Step 1:
Determine the optimal decision and its expectedDetermine the optimal decision and its expectedreturn for the possible outcomes of the sample usingreturn for the possible outcomes of the sample usingthe posterior probabilities for the states of nature.the posterior probabilities for the states of nature.
•• Step 2:Step 2:Compute the expected value of these optimalCompute the expected value of these optimal
returns.returns.•• Step 3:Step 3:
Subtract the EV of the optimal decision obtainedSubtract the EV of the optimal decision obtainedwithout using the sample information from thewithout using the sample information from theamount determined in step (2).amount determined in step (2).
Efficiency of Sample InformationEfficiency of Sample Information
Efficiency of sample informationEfficiency of sample information is the ratio of EVSIis the ratio of EVSIto EVPI.to EVPI.
As the EVPI provides an upper bound for the EVSI,As the EVPI provides an upper bound for the EVSI,efficiency is always a number between 0 and 1.efficiency is always a number between 0 and 1.
1515
Burger Prince must decide whether or not toBurger Prince must decide whether or not topurchase a marketing survey from Stanton Marketingpurchase a marketing survey from Stanton Marketingfor $1,000. The results of the survey are "favorable" orfor $1,000. The results of the survey are "favorable" or"unfavorable". The conditional probabilities are:"unfavorable". The conditional probabilities are:
P(favorable | 80 customers per hour) = .2P(favorable | 80 customers per hour) = .2P(favorable | 100 customers per hour) = .5P(favorable | 100 customers per hour) = .5P(favorable | 120 customers per hour) = .9P(favorable | 120 customers per hour) = .9
Should Burger Prince have the survey performedShould Burger Prince have the survey performedby Stanton Marketing?by Stanton Marketing?
Sample InformationSample Information
Influence DiagramInfluence Diagram
RestaurantRestaurantSizeSize ProfitProfit
Avg. NumberAvg. Numberof Customersof Customers
Per HourPer Hour
MarketMarketSurveySurveyResultsResults
MarketMarketSurveySurvey
DecisionDecisionChanceChanceConsequenceConsequence
FavorableFavorable
StateState PriorPrior ConditionalConditional JointJoint PosteriorPosterior80 .4 .2 .08 .14880 .4 .2 .08 .148
100 .2 .5 .10 .185100 .2 .5 .10 .185120 .4 .9120 .4 .9 .36.36 .667.667
Total .54 1.000Total .54 1.000
P(favorable) = .54P(favorable) = .54
Posterior ProbabilitiesPosterior Probabilities
1616
UnfavorableUnfavorable
StateState PriorPrior ConditionalConditional JointJoint PosteriorPosterior80 .4 .8 .32 .69680 .4 .8 .32 .696
100 .2 .5 .10 .217100 .2 .5 .10 .217120 .4 .1120 .4 .1 .04.04 .087.087
Total .46 1.000Total .46 1.000
P(unfavorable) = .46P(unfavorable) = .46
Posterior ProbabilitiesPosterior Probabilities
Formula SpreadsheetFormula SpreadsheetA B C D E
12 Prior Conditional Joint Posterior3 State of Nature Probabilities Probabilities Probabilities Probabilities4 s1 = 80 0.4 0.2 =B4*C4 =D4/$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) =
Posterior ProbabilitiesPosterior Probabilities
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) =
Posterior ProbabilitiesPosterior Probabilities
1717
Decision TreeDecision Tree
Top HalfTop Half
ss11 (.148)(.148)
ss11 (.148)(.148)
ss11 (.148)(.148)ss22 (.185)(.185)
ss22 (.185)(.185)
ss22 (.185)(.185)
ss33 (.667)(.667)
ss33 (.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
Bottom HalfBottom Half
ss11 (.696)(.696)
ss11 (.696)(.696)
ss11 (.696)(.696)
ss22 (.217)(.217)
ss22 (.217)(.217)
ss22 (.217)(.217)
ss33 (.087)(.087)
ss33 (.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
Decision TreeDecision Tree
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
Decision TreeDecision Tree
1818
If the outcome of the survey is "favorable”,If the outcome of the survey is "favorable”,choose Model C. If it is “unfavorable”, choose Model A.choose Model C. If it is “unfavorable”, choose Model A.
EVSI = .54($17,855) + .46($11,433)EVSI = .54($17,855) + .46($11,433) -- $14,000 = $900.88$14,000 = $900.88
Since this is less than the cost of the survey, theSince this is less than the cost of the survey, thesurvey shouldsurvey should notnot be purchased.be purchased.
Expected Value of Sample InformationExpected Value of Sample Information
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