1 decision-making under uncertainty lesson 12 lecture twelve decision -making under uncertainity

1 Decision-Making Under Uncertainty

Lesson 12

LECTURE TWELVELECTURE TWELVE

DecisionDecision-Making-Making

UNDER UNDER UNCERTAINITYUNCERTAINITY


Lesson 12

IntroductionIntroductionDecision analysis provides a framework and Decision analysis provides a framework and

methodology for rational decision making when methodology for rational decision making when the outcomes are uncertain.the outcomes are uncertain.

ExampleExample: A company plans to determine the best : A company plans to determine the best location to startup a new plant from a choice of location to startup a new plant from a choice of several locations. Each location offers a different several locations. Each location offers a different cost scenario. Decision Analysis techniques can cost scenario. Decision Analysis techniques can be used to determine the best decision. be used to determine the best decision.


Lesson 12

Payoff TablePayoff Table

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 -3 1 5 -3

States of Nature refer to future events which may occur States of Nature refer to future events which may occur and the values in a Payoff Table refer to either Profit or and the values in a Payoff Table refer to either Profit or Cost. Example: s1, s2 & s3 could refer to possible Cost. Example: s1, s2 & s3 could refer to possible scenarios (i.e. Moderate, Strong & Weak market scenarios (i.e. Moderate, Strong & Weak market demand)demand)


Lesson 12

Simple Decision Making Simple Decision Making without Probabilitieswithout Probabilities

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

– the the OptimisticOptimistic approach (Maximax or Minimin) approach (Maximax or Minimin)

– the the ConservativeConservative approach (Maximin or Minimax) approach (Maximin or Minimax)

– the the Minimax RegretMinimax Regret approach. approach.


Lesson 12

Optimistic ApproachOptimistic Approach• The The optimistic approachoptimistic approach would be used would be used

by an optimistic decision maker.by an optimistic decision maker.

• The The decision with the decision with the largest possible largest possible payoffpayoff is chosen. is chosen.

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


Lesson 12

Example: Optimistic ApproachExample: Optimistic ApproachConsider the following problem with three Consider the following problem with three

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


ss11 ss22 ss33

dd11 4 4 -2 4 4 -2


dd33 1 5 -3 1 5 -3


Lesson 12

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

The Optimistic Approach is to make decision based on theThe Optimistic Approach is to make decision based on themaximum of the largest profits. For each decision, d, maximum of the largest profits. For each decision, d, the largest profits are identified (i.e. 4, -1 & 5)the largest profits are identified (i.e. 4, -1 & 5)


Lesson 12

Example: Optimistic ApproachExample: Optimistic Approach

• Solution SpreadsheetSolution Spreadsheet

A B C D E F123 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


Lesson 12

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 For each decision the minimum payoffminimum payoff is listed and is listed and then the decision corresponding to the maximum then the decision corresponding to the maximum of these minimum payoffs is selected. of these minimum payoffs is selected.

• 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. these maximum costs is selected.


Lesson 12

Example: Conservative ApproachExample: Conservative ApproachBased on the same table with three decision alternatives and Based on the same table with three decision alternatives and three states of nature with the following payoff table three states of nature with the following payoff table representing profits:representing profits:


ss11 ss22 ss33

dd11 4 4 -2 4 4 -2


dd33 1 5 -3 1 5 -3


Lesson 12

Example: Conservative ApproachExample: Conservative Approach• 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


Lesson 12

Example: Conservative ApproachExample: Conservative Approach

• 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


Lesson 12

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 opportunity loss tabletable. .

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


Lesson 12

Compute a regret table by subtracting each payoff in a column from the largest payoff in that column. Add a Max Regret Col and make decision based on the Minimum of Maximum Regret. Example: 4, 0, 1 is Example: 4, 0, 1 is subtracted by 4 giving OL 0, 4, 3, etc.subtracted by 4 giving OL 0, 4, 3, etc.

s1 OLOL s2 OLOL s3 OL Max RegretOL Max Regret

d1 4 00 4 11 -2 11 1 1

d2 0 44 3 22 -1 00 44

d3 1 33 5 0 0 -3 2 32 3

Example: Minimax Regret ApproachExample: Minimax Regret Approach


Lesson 12

Example: Minimax Regret Approach Example: Minimax Regret Approach • Formula SpreadsheetFormula Spreadsheet

A B C D E F12 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


Lesson 12

• Solution Spreadsheet

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



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


Lesson 12

Expected Value ApproachExpected Value Approach– If probabilities regarding the states of nature is available, we If probabilities regarding the states of nature is available, we

may use the expected value (EV) approach. Decision is may use the expected value (EV) approach. Decision is based on maximising EV.based on maximising EV.

– The expected value (EV) of decision alternative The expected value (EV) of decision alternative ddii is defined is defined as: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 decision = the payoff corresponding to decision alternative alternative ddii and state of nature and state of nature ssjj

Decision Making with ProbabilitiesDecision Making with Probabilities

EV( ) ( )d P s Vi j ijj

N

1

EV( ) ( )d P s Vi j ijj

N

1


Lesson 12

ABC RestaurantABC Restaurant

Average Number of Customers Per HourAverage Number of Customers Per Hour

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

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

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

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

ProbabilityProbability 0.4 0.4 0.20.2 0.4 0.4

Given s1, s2 & s3 have the probabilities: 0.4, 0.2 Given s1, s2 & s3 have the probabilities: 0.4, 0.2 & 0.4 & 0.4

Example: Expected Value ApproachExample: Expected Value Approach


Lesson 12


• Formula 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


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


Lesson 12

• Solution 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



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



Lesson 12

Expected Value of Perfect InformationExpected Value of Perfect Information

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

• EVPI = EVwPI – Max EVEVPI = EVwPI – Max EV

where EVwPI = ∑ Pi * Max Viwhere EVwPI = ∑ Pi * Max Viwhere Vi is the payoffswhere Vi is the payoffs

and EVwPI = Expected Value with Perfect and EVwPI = Expected Value with Perfect Information Information


Lesson 12

• Spreadsheet

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

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

Expected Value of Perfect InformationExpected Value of Perfect Information


Lesson 12

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

Average Number of Customers Per HourAverage Number of Customers Per Hour ss1 = 80 1 = 80 ss2 = 100 2 = 100 ss3 = 1203 = 120Model 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 $21,000ProbabilitiesProbabilities 0.4 0.4 0.2 0.40.2 0.4


Lesson 12

Decision TreeDecision Tree

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


Lesson 12

Decision Tree with Tree PlanDecision Tree with Tree Plan

Open ‘tree164e.xla’ and enable Macro first.Open ‘tree164e.xla’ and enable Macro first.


Lesson 12

Decision Tree with Tree Decision Tree with Tree PlanPlan


Lesson 12

0.4Event 4

1000010000 10000

0.2Decision 1 Event 5

150000 12600 15000 15000

0.4Event 6

1400014000 14000

0.4Event 7

80008000 8000


3 1800014000 0 11600 18000 18000

0.4Event 9

1200012000 12000

0.4Event 10

60006000 6000


160000 14000 16000 16000

0.4Event 12

2100021000 21000

Decision Decision Tree with Tree with Tree PlanTree Plan


Lesson 12

TOM BROWN INVESTMENT TOM BROWN INVESTMENT DECISIONDECISION

• Tom Brown has inherited $1000.• He has to decide how to invest the money for one

year.• A broker has suggested five potential

investments.– Gold– Junk Bond– Growth Stock– Certificate of Deposit– Stock Option Hedge


Lesson 12

• The return on each investment depends on the (uncertain) market behavior during the year.

• Tom would build a payoff table to help make the investment decision

TOM BROWNTOM BROWN


Lesson 12

Alternatives Large Rise Small Rise No Change Small Fall Large Fall

Gold -100 100 200 300 0Bond 250 200 150 -100 -150Stock 500 250 100 -200 -600C/D account 60 60 60 60 60Stock option 200 150 150 -200 -150

Probability 0.2 0.3 0.3 0.1 0.1

The Payoff TableThe Payoff TableDJA is down more DJA is down more than 800 pointsthan 800 points

DJA is down DJA is down [-300, -800][-300, -800]

DJA movesDJA moveswithin within [-300,+300] [-300,+300]

DJA is up DJA is up [+300,[+300,++1000] 1000]

DJA is up more DJA is up more than1000 pointsthan1000 points


Lesson 12

TaskTask

Evaluate each investment alternative using:• Maximax approach• Maximin approach• Minimax Regret approach• EV approach• And construct Decision Tree for EV approach• Calculate EVPI


Lesson 12

QUESTIONS


Lesson 12

Review Questions:Review Questions:

1. What is meant by ‘decision-making under uncertainty’?

2. Why should sequential decisions be considered differently from a series of separate decisions?

3. How can you identify the best decisions in a decision tree?

1 decision-making under uncertainty lesson 12 lecture twelve decision -making under uncertainity

Documents

best decision

decision corresponding

optimistic decision

decision alternatives

rational decision

conservative decision

simple decision making

uncertainty lesson