Decision Analysis

11©© 2003 2003 ThomsonThomsonΤΜΤΜ/South/South--WesternWestern SlideSlide

Chapter 1 Decision AnalysisChapter 1 Decision Analysis

Problem FormulationProblem FormulationDecision Making without ProbabilitiesDecision Making without ProbabilitiesDecision Making with ProbabilitiesDecision Making with ProbabilitiesRisk Analysis and Sensitivity AnalysisRisk Analysis and Sensitivity AnalysisDecision Analysis with Sample InformationDecision Analysis with Sample InformationComputing Branch ProbabilitiesComputing Branch ProbabilitiesUtility and Decision MakingUtility and Decision Making


Problem FormulationProblem Formulation

A decision problem is characterized by decision A decision problem is characterized by decision alternatives, states of nature, and resulting payoffs.alternatives, states of nature, and resulting payoffs.The The decision alternativesdecision alternatives are the different possible are the different possible strategies the decision maker can employ.strategies the decision maker can employ.The The states of naturestates of nature refer to future events, not under refer to future events, not under the control of the decision maker, which may occur. the control of the decision maker, which may occur. States of nature should be defined so that they are States of nature should be defined so that they are mutually exclusive and collectively exhaustive.mutually exclusive and collectively exhaustive.


Influence DiagramsInfluence Diagrams

An An influence diagraminfluence diagram is a graphical device showing the is a graphical device showing the relationships among the decisions, the chance events, relationships among the decisions, the chance events, and the consequences.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 direction of connecting the nodes show the direction of influence.influence.


Payoff TablesPayoff Tables

The consequence resulting from a specific combination The consequence resulting from a specific combination of a decision alternative and a state of nature is a of a decision alternative and a state of nature is a payoffpayoff..A table showing payoffs for all combinations of A table showing payoffs for all combinations of decision alternatives and states of nature is a decision alternatives and states of nature is a payoff payoff tabletable..Payoffs can be expressed in terms of Payoffs can be expressed in terms of profitprofit, , costcost, , timetime, , distancedistance or any other appropriate measure.or any other appropriate measure.


Decision TreesDecision Trees

A A decision treedecision tree is a chronological representation of is a chronological representation of the decision problem.the decision problem.Each decision tree has two types of nodes; Each decision tree has two types of nodes; round round nodesnodes correspond to the states of nature while correspond to the states of nature while square square nodesnodes correspond to the decision alternatives. correspond to the decision alternatives. The The branchesbranches leaving each round node represent the leaving each round node represent the different states of nature while the branches leaving different states of nature while the branches leaving each square node represent the different decision each square node represent the different decision alternatives.alternatives.At the end of each limb of a tree are the payoffs At the end of each limb of a tree are the payoffs attained from the series of branches making up that attained from the series of branches making up that limb. limb.


Decision Making without ProbabilitiesDecision Making without Probabilities

Three commonly used criteria for decision making Three commonly used criteria for decision making when probability information regarding the when probability information regarding the likelihood of the states of nature is unavailable are: likelihood of the states of nature is unavailable are: ••the the optimisticoptimistic approachapproach••the the conservativeconservative approachapproach••the the minimaxminimax regretregret approach. approach.


Optimistic ApproachOptimistic Approach

The The optimistic approachoptimistic approach would be used by an would be used by an optimistic decision maker.optimistic decision maker.The The decision with the largest possible payoffdecision with the largest possible payoff is is chosen. chosen. If the payoff table was in terms of costs, the If the payoff table was in terms of costs, the decision decision with the lowest costwith the lowest cost would be chosen.would be chosen.


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 and For each decision the minimum payoff is listed and then the decision corresponding to the maximum of then the decision corresponding to the maximum of these minimum payoffs is selected. (Hence, the these minimum payoffs is selected. (Hence, the minimum possible payoff is maximizedminimum possible payoff is maximized.).)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 of then the decision corresponding to the minimum of these maximum costs is selected. (Hence, the these maximum costs is selected. (Hence, the maximum possible cost is minimizedmaximum possible cost is minimized.).)


MinimaxMinimax Regret ApproachRegret Approach

The The minimaxminimax regret approach requires the regret approach requires the construction of a construction of a regret tableregret table or an or an opportunity loss opportunity loss tabletable. . This is done by calculating for each state of nature This is done by calculating for each state of nature the difference between each payoff and the largest the difference between each payoff and the largest payoff for that state of nature. payoff for that state of nature. Then, using this regret table, the maximum regret for Then, using this regret table, the maximum regret for each possible decision is listed. each possible decision is listed. The decision chosen is the one corresponding to the The decision chosen is the one corresponding to the minimum of the maximum regretsminimum of the maximum regrets..


ExampleExample

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

States of NatureStates of Naturess11 ss22 ss33

dd11 4 4 4 4 --22DecisionsDecisions dd22 0 3 0 3 --11

dd33 1 5 1 5 --33


ExampleExample

Optimistic ApproachOptimistic ApproachAn optimistic decision maker would use the An optimistic decision maker would use the

optimistic (optimistic (maximaxmaximax) approach. We choose the ) approach. We choose the decision that has the largest single value in the payoff decision that has the largest single value in the payoff table. table.

MaximumMaximumDecisionDecision PayoffPayoffdd11 44dd22 33dd33 55

MaximaxMaximaxpayoffpayoff

MaximaxMaximaxdecisiondecision


ExampleExample

Formula Spreadsheet for Optimistic ApproachFormula Spreadsheet for Optimistic ApproachA 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


ExampleExample

Solution Spreadsheet for Optimistic ApproachSolution Spreadsheet for Optimistic ApproachA 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


ExampleExample

Conservative ApproachConservative ApproachA conservative decision maker would use the A conservative decision maker would use the

conservative (conservative (maximinmaximin) approach. List the minimum ) approach. List the minimum payoff for each decision. Choose the decision with the payoff for each decision. Choose the decision with the maximum of these minimum payoffs.maximum of these minimum payoffs.

MinimumMinimumDecisionDecision PayoffPayoffdd11 --22dd22 --11dd33 --33

MaximinMaximindecisiondecision

MaximinMaximinpayoffpayoff


ExampleExample

Formula Spreadsheet for Conservative ApproachFormula Spreadsheet for Conservative ApproachA 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


ExampleExample

Solution Spreadsheet for Conservative ApproachSolution Spreadsheet for Conservative Approach

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


MinimaxMinimax Regret ApproachRegret ApproachFor the For the minimaxminimax regret approach, first compute a regret approach, first compute a

regret table by subtracting each payoff in a column regret table by subtracting each payoff in a column from the largest payoff in that column. In this from the largest payoff in that column. In this example, in the first column subtract 4, 0, and 1 from example, in the first column subtract 4, 0, and 1 from 4; 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

ExampleExample


ExampleExample

MinimaxMinimax Regret Approach (continued)Regret Approach (continued)For each decision list the maximum regret. Choose For each decision list the maximum regret. Choose

the decision with the minimum of these values.the decision with the minimum of these values.

MaximumMaximumDecisionDecision RegretRegret

dd11 11dd22 44dd33 3 3

MinimaxMinimaxdecisiondecision

MinimaxMinimaxregretregret


ExampleExample

Formula Spreadsheet for Formula Spreadsheet for MinimaxMinimax Regret ApproachRegret ApproachA B C D E F

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


ExampleExample

Solution Spreadsheet for Solution Spreadsheet for MinimaxMinimax Regret ApproachRegret ApproachA B C D E F

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




Decision Making with ProbabilitiesDecision Making with Probabilities

Expected Value ApproachExpected Value Approach••If probabilistic information regarding the states of If probabilistic information regarding the states of

nature is available, one may use the nature is available, one may use the expected expected value (EV) approachvalue (EV) approach. .

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

••The decision yielding the The decision yielding the best expected returnbest expected return is is chosen.chosen.


Expected Value of a Decision AlternativeExpected Value of a Decision Alternative

The The expected value of a decision alternativeexpected value of a decision alternative is the sum is the sum of weighted payoffs for the decision alternative.of weighted payoffs for the decision alternative.The expected value (EV) of decision alternative The expected value (EV) of decision alternative ddii is is defined 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

alternative alternative ddii and state of nature and state of nature ssjj

EV( ) ( )d P s Vi j ijj

N

==∑

1EV( ) ( )d P s Vi j ij

j

N

==∑

1


Example: Burger PrinceExample: Burger Prince

Burger Prince Restaurant is contemplating Burger Prince Restaurant is contemplating opening a new restaurant on Main Street. It has three opening a new restaurant on Main Street. It has three different models, each with a different seating different models, each with a different seating capacity. Burger Prince estimates that the average capacity. Burger Prince estimates that the average number of customers per hour will be 80, 100, or 120. number of customers per hour will be 80, 100, or 120. The payoff table for the three models is on the next The payoff table for the three models is on the next slide. 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



Expected Value ApproachExpected Value ApproachCalculate the expected value for each decision. The Calculate the expected value for each decision. The

decision tree on the next slide can assist in this decision tree on the next slide can assist in this calculation. Here calculation. Here dd11, , dd22, , dd3 3 represent the decision represent the decision alternatives of models A, B, C, and alternatives of models A, B, C, and ss11, , ss22, , ss3 3 represent represent the states of nature of 80, 100, and 120.the states of nature of 80, 100, and 120.



Decision TreeDecision Tree

111

.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

PayoffsPayoffs

10,00010,000

15,00015,000

14,00014,0008,0008,000

18,00018,00012,00012,0006,0006,000

16,00016,00021,00021,000

222

333

444



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.

333

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

222

111

444



Formula Spreadsheet for Expected Value ApproachFormula Spreadsheet for Expected Value ApproachA 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


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


Solution Spreadsheet for Expected Value ApproachSolution Spreadsheet for Expected Value Approach


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 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


Expected Value of Perfect InformationExpected Value of Perfect Information

Frequently information is available which can Frequently information is available which can improve the probability estimates for the states of improve the probability estimates for the states of nature. nature. The The expected value of perfect informationexpected value of perfect information (EVPI) is (EVPI) is the increase in the expected profit that would result if the increase in the expected profit that would result if one knew with certainty which state of nature would one knew with certainty which state of nature would occur. occur. The EVPI provides an The EVPI provides an upper bound on the expected upper bound on the expected value of any sample or survey informationvalue of any sample or survey information. .


Expected Value of Perfect InformationExpected Value of Perfect Information

EVPI CalculationEVPI Calculation••Step 1:Step 1:

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

••Step 2:Step 2:Compute the expected value of these optimal Compute the expected value of these optimal

returns.returns.••Step 3:Step 3:

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



Expected Value of Perfect InformationExpected Value of Perfect InformationCalculate the expected value for the optimum Calculate the expected value for the optimum

payoff for each state of nature and subtract the EV of payoff for each state of nature and subtract the EV of the 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



Spreadsheet for Expected Value of Perfect InformationSpreadsheet for Expected Value of Perfect InformationA 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


Risk AnalysisRisk Analysis

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

The The risk profilerisk profile for a decision alternative shows the for a decision alternative shows the possible payoffs for the decision alternative along with possible payoffs for the decision alternative along with their associated probabilities.their associated probabilities.



Risk Profile for the Model C Decision AlternativeRisk Profile for the Model 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

Profit ($thousands)Profit ($thousands)


Sensitivity AnalysisSensitivity Analysis

Sensitivity analysisSensitivity analysis can be used to determine how can be used to determine how changes to the following inputs affect the 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 inputs If a small change in the value of one of the inputs causes a change in the recommended decision causes a change in the recommended decision alternative, extra effort and care should be taken in alternative, extra effort and care should be taken in estimating the input value.estimating the input value.


BayesBayes’’ Theorem and Posterior ProbabilitiesTheorem and Posterior Probabilities

Knowledge of sample or survey information can be Knowledge of sample or survey information can be used to revise the probability estimates for the states of used to revise the probability estimates for the states of nature. 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 revised by information, these prior probabilities can be revised by employing employing BayesBayes' Theorem' 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 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 prior For each state of nature, multiply the prior probability by its conditional probability for the probability by its conditional probability for the indicator indicator ---- this gives the this gives the joint probabilitiesjoint probabilities for the for the states and indicator.states and indicator.

••Step 2:Step 2:Sum these joint probabilities over all states Sum these joint probabilities over all states ---- this this

gives the gives the marginal probabilitymarginal probability for the indicator.for the indicator.••Step 3:Step 3:

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


Expected Value of Sample InformationExpected Value of Sample Information

The The expected value of sample informationexpected value of sample information (EVSI) is (EVSI) is the additional expected profit possible through the additional expected profit possible through knowledge 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 expected Determine the optimal decision and its expected return for the possible outcomes of the sample using return for the possible outcomes of the sample using the 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 optimal Compute the expected value of these optimal

returns.returns.••Step 3:Step 3:

Subtract the EV of the optimal decision obtained Subtract the EV of the optimal decision obtained without using the sample information from the without using the sample information from the amount 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 EVSI to is the ratio of EVSI to EVPI. 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.


Sample InformationSample InformationBurger Prince must decide whether or not to Burger Prince must decide whether or not to

purchase a marketing survey from Stanton Marketing purchase a marketing survey from Stanton Marketing for $1,000. The results of the survey are "favorable" or for $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) = .5 P(favorable | 100 customers per hour) = .5 P(favorable | 120 customers per hour) = .9 P(favorable | 120 customers per hour) = .9

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




Influence DiagramInfluence Diagram

RestaurantRestaurantSizeSize ProfitProfit

Avg. NumberAvg. Numberof Customersof Customers

Per HourPer Hour

MarketMarketSurveySurveyResultsResults

MarketMarketSurveySurvey

DecisionDecisionChanceChanceConsequenceConsequence



Posterior ProbabilitiesPosterior Probabilities

FavorableFavorable

StateState PriorPrior ConditionalConditional JointJoint PosteriorPosterior80 .4 .2 .08 80 .4 .2 .08 .148.148

100 .2 .5 .10 100 .2 .5 .10 .185.185120 .4 .9 120 .4 .9 .36.36 .667.667

Total .54 Total .54 1.0001.000

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



Posterior ProbabilitiesPosterior Probabilities

UnfavorableUnfavorable

StateState PriorPrior ConditionalConditional JointJoint PosteriorPosterior80 .4 .8 .32 80 .4 .8 .32 .696.696

100 .2 .5 .10 100 .2 .5 .10 .217.217120 .4 .1 120 .4 .1 .04.04 .087.087

Total .46 1.000Total .46 1.000

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



Formula Spreadsheet for Posterior ProbabilitiesFormula Spreadsheet for Posterior ProbabilitiesA 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) =



Solution Spreadsheet for Posterior ProbabilitiesSolution Spreadsheet for Posterior ProbabilitiesA 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) =



Decision Tree (top half)Decision Tree (top 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

222

444

555

666

111



Decision Tree (bottom half)Decision Tree (bottom 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

777

999

888333

111



II22(.46)(.46)

dd11

dd22

dd33

EMV = .696(10,000) + .217(15,000)EMV = .696(10,000) + .217(15,000)+.087(14,000)= $11,433+.087(14,000)= $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) = $17,855+.667(21,000) = $17,855

444

555

666

777

888

999

222

333

111

$17,855$17,855

$11,433$11,433


Expected Value of Sample InformationExpected Value of Sample InformationIf the outcome of the survey is "favorableIf the outcome of the survey is "favorable””, choose , choose

Model C. If it is Model C. If it is ““unfavorableunfavorable””, choose model A., 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, the Since this is less than the cost of the survey, the survey should survey should notnot be purchased.be purchased.




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


Meaning of UtilityMeaning of Utility

Utilities are used when the decision criteria must be based on more than just expected monetary values.Utility is a measure of the total worth of a particular outcome, reflecting the decision maker’s attitude towards a collection of factors. Some of these factors may be profit, loss, and risk.This analysis is particularly appropriate in cases where payoffs can assume extremely high or extremely low values.


Weiss Advisors have analyzed the profit potential Weiss Advisors have analyzed the profit potential of five different investments. The probabilities of the of five different investments. The probabilities of the gains on $1000 are as follows:gains on $1000 are as follows:

GainGainInvestmentInvestment $0$0 $200$200 $500$500 $1000$1000

A A .9 0 0 .1.9 0 0 .1B B 0 .8 .2 00 .8 .2 0C C .05 .9 0 .05.05 .9 0 .05D 0 .8 .1 .1D 0 .8 .1 .1E E .6 0 .3 .1.6 0 .3 .1

One of WeissOne of Weiss’’ investors informs Weiss that he is investors informs Weiss that he is indifferent between investments A, B, and C.indifferent between investments A, B, and C.

Example: Weiss AdvisorsExample: Weiss Advisors



Developing Utilities for PayoffsDeveloping Utilities for Payoffs

Assign a utility of 10 to a $1000 gain and a utility of Assign a utility of 10 to a $1000 gain and a utility of 0 to a gain of $0. Let 0 to a gain of $0. Let xx = the utility of a $200 gain and = the utility of a $200 gain and yy = the utility of a $500 gain. The expected utility on = the utility of a $500 gain. The expected utility on investment A is then .9(0) + .1(10) = 1. investment A is then .9(0) + .1(10) = 1.

Since the investor is indifferent between investments Since the investor is indifferent between investments A and C, this must mean the expected utility of A and C, this must mean the expected utility of investment C = the expected utility of investment A = 1. investment C = the expected utility of investment A = 1. But the expected utility of investment C = .05(0) +.90But the expected utility of investment C = .05(0) +.90xx + + .05(10). Since this must equal 1, solving for .05(10). Since this must equal 1, solving for xx, gives , gives xx = = 5/9.5/9.



Developing Utilities for Payoffs (continue)Developing Utilities for Payoffs (continue)

Also since the investor is indifferent between A, B, Also since the investor is indifferent between A, B, and C, the expected utility of investment B must be 1. and C, the expected utility of investment B must be 1. Thus, 0(0) + .8(5/9) + .2Thus, 0(0) + .8(5/9) + .2yy + 0(10) = 1. Solving for + 0(10) = 1. Solving for yy, , gives gives yy = 25/9. = 25/9.

Thus the utility values for gains of 0, 200, 500, and Thus the utility values for gains of 0, 200, 500, and 1000 are 0, 5/9, 25/9, and 10, respectively.1000 are 0, 5/9, 25/9, and 10, respectively.


Expected Utility ApproachExpected Utility Approach

Once a utility function has been determined, the optimal decision can be chosen using the expected utility approach. Here, the expected utility for each decision alternative is computed as:

The decision alternative with the highest expected utility is chosen.

=

= ∑1

EU( ) ( )N

i j ijj

d P s U=

= ∑1

EU( ) ( )N

i j ijj

d P s U


Example: Risk AvoiderExample: Risk Avoider

Consider a threeConsider a three--state, threestate, three--decision problem with decision problem with the following payoff table in dollars:the following payoff table in dollars:

ss11 ss22 ss33

dd11 +100,000+100,000 +40,000+40,000 --60,00060,000dd22 +50,000+50,000 +20,000+20,000 --30,00030,000dd33 +20,000+20,000 +20,000+20,000 --10,00010,000

The probabilities for the three states of nature are: The probabilities for the three states of nature are: P(P(ss11) = .1, P() = .1, P(ss22) = .3, and P() = .3, and P(ss33) = .6.) = .6.



Utility Table for Decision MakerUtility Table for Decision Maker

ss11 ss22 ss33

dd1 1 100100 9090 00dd22 9494 8080 4040dd33 8080 8080 6060


Expected Utility

ExpectedExpectedss11 ss22 ss33 UtilityUtility

dd11 100100 9090 00 37.037.0dd22 9494 8080 4040 57.457.4dd33 8080 8080 6060 68.068.0

Probability .1Probability .1 .3.3 .6.6

Decision maker should choose decision Decision maker should choose decision dd33..



End of Chapter 1End of Chapter 1

Decision Analysis

Documents

Transcript of Decision Analysis