1 6.1 introduction to decision analysis the field of decision analysis provides a framework for...
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6.1 Introduction to Decision Analysis6.1 Introduction to Decision Analysis
• The field of decision analysis provides a framework for making important decisions.
• Decision analysis allows us to select a decision from a set of possible decision alternatives when uncertainties regarding the future exist.
• The goal is to optimize the resulting payoff in terms of a decision criterion.
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• Maximizing the decision maker’s utility
function is the mechanism used when risk
is factored into the decision making
process.
• Maximizing expected profit is a common
criterion when probabilities can be
assessed.
6.1 Introduction to Decision Analysis6.1 Introduction to Decision Analysis
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6.26.2 Payoff Table Analysis Payoff Table Analysis
• Payoff Tables
– Payoff table analysis can be applied when:• There is a finite set of discrete decision alternatives.• The outcome of a decision is a function of a single future event.
– In a Payoff table -• The rows correspond to the possible decision alternatives.• The columns correspond to the possible future events.• Events (states of nature) are mutually exclusive and collectively
exhaustive.• The table entries are the payoffs.
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TOM BROWN INVESTMENT DECISIONTOM BROWN INVESTMENT DECISION
• 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
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• 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
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S1 S2 S3 S4
D1 p11 p12 p13 p14
D2 p21 p22 p23 P24
D3 p31 p32 p33 p34
• Select a decision making criterion, and apply it to the payoff table.
TOM BROWN - SolutionTOM BROWN - Solution
S1 S2 S3 S4
D1 p11 p12 p13 p14
D2 p21 p22 p23 P24
D3 p31 p32 p33 p34
Criterion
P1P2P3
• Construct a payoff table.
• Identify the optimal decision.
• Evaluate the solution.
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Decision States of Nature
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
The Payoff TableThe Payoff Table
The states of nature are mutually exclusive and collectively exhaustive.
Define the states of nature.
DJA is down more than 800 points
DJA is down [-300, -800]
DJA moveswithin [-300,+300]
DJA is up [+300,+1000]
DJA is up more than1000 points
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Decision States of Nature
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
The Payoff TableThe Payoff Table
Determine the set of possible decision alternatives.
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Decision States of Nature
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
The stock option alternative is dominated by the bond alternative
250 200 150 -100 -150
-150
The Payoff TableThe Payoff Table
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6.3 Decision Making Criteria6.3 Decision Making Criteria
• Classifying decision-making criteria
– Decision making under certainty.• The future state-of-nature is assumed known.
– Decision making under risk.• There is some knowledge of the probability of the states of
nature occurring.– Decision making under uncertainty.
• There is no knowledge about the probability of the states of nature occurring.
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• The decision criteria are based on the decision maker’s attitude toward life.
• The criteria include the– Maximin Criterion - pessimistic or conservative approach.– Minimax Regret Criterion - pessimistic or conservative approach.– Maximax Criterion - optimistic or aggressive approach.– Principle of Insufficient Reasoning – no information about the
likelihood of the various states of nature.
Decision Making Under UncertaintyDecision Making Under Uncertainty
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Decision Making Under Uncertainty - Decision Making Under Uncertainty - The Maximin CriterionThe Maximin Criterion
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• This criterion is based on the worst-case scenario. – It fits both a pessimistic and a conservative decision
maker’s styles.
– A pessimistic decision maker believes that the worst
possible result will always occur.
– A conservative decision maker wishes to ensure a guaranteed minimum possible payoff.
Decision Making Under Uncertainty - Decision Making Under Uncertainty - The Maximin CriterionThe Maximin Criterion
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TOM BROWN - The Maximin CriterionTOM BROWN - The Maximin Criterion
• To find an optimal decision
– Record the minimum payoff across all states of nature for
each decision.
– Identify the decision with the maximum “minimum payoff.”
The Maximin Criterion Minimum
Decisions Large Rise Small rise No Change Small Fall Large Fall Payoff
Gold -100 100 200 300 0 -100Bond 250 200 150 -100 -150 -150Stock 500 250 100 -200 -600 -600C/D account 60 60 60 60 60 60
The Maximin Criterion Minimum
Decisions Large Rise Small rise No Change Small Fall Large Fall Payoff
Gold -100 100 200 300 0 -100Bond 250 200 150 -100 -150 -150Stock 500 250 100 -200 -600 -600C/D account 60 60 60 60 60 60
The optimal decision
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=MAX(H4:H7)
* FALSE is the range lookup argument in the VLOOKUP function in cell B11 since the values in column H are not in ascending order
=VLOOKUP(MAX(H4:H7),H4:I7,2,FALSE)
=MIN(B4:F4)Drag to H7
The Maximin Criterion - spreadsheetThe Maximin Criterion - spreadsheet
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To enable the spreadsheet to correctly identify the optimal maximin decision in cell B11, the labels for cells A4 through A7 are copied into cells I4 through I7 (note that column I in the spreadsheet is hidden).
I4
Cell I4 (hidden)=A4Drag to I7
The Maximin Criterion - spreadsheetThe Maximin Criterion - spreadsheet
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Decision Making Under Uncertainty - Decision Making Under Uncertainty - The Minimax Regret CriterionThe Minimax Regret Criterion
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• The Minimax Regret Criterion– This criterion fits both a pessimistic and a
conservative decision maker approach.– The payoff table is based on “lost opportunity,” or
“regret.”– The decision maker incurs regret by failing to choose
the “best” decision.
Decision Making Under Uncertainty - Decision Making Under Uncertainty - The Minimax Regret CriterionThe Minimax Regret Criterion
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• The Minimax Regret Criterion– To find an optimal decision, for each state of nature:
• Determine the best payoff over all decisions.• Calculate the regret for each decision alternative as the
difference between its payoff value and this best payoff value.
– For each decision find the maximum regret over all states of nature.
– Select the decision alternative that has the minimum of these “maximum regrets.”
Decision Making Under Uncertainty - Decision Making Under Uncertainty - The Minimax Regret CriterionThe Minimax Regret Criterion
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• This criterion is based on the best possible scenario.It fits both an optimistic and an aggressive decision maker.
• An optimistic decision maker believes that the best possible outcome will always take place regardless of the decision made.
• An aggressive decision maker looks for the decision with the highest payoff (when payoff is profit).
Decision Making Under Uncertainty - Decision Making Under Uncertainty - The Maximax CriterionThe Maximax Criterion
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• To find an optimal decision.– Find the maximum payoff for each decision
alternative.– Select the decision alternative that has the maximum
of the “maximum” payoff.
Decision Making Under Uncertainty - Decision Making Under Uncertainty - The Maximax CriterionThe Maximax Criterion
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TOM BROWN -TOM BROWN - The Maximax CriterionThe Maximax Criterion
The Maximax Criterion MaximumDecision Large rise Small rise No change Small fall Large fall PayoffGold -100 100 200 300 0 300Bond 250 200 150 -100 -150 200Stock 500 250 100 -200 -600 500C/D 60 60 60 60 60 60
The optimal decision
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• This criterion might appeal to a decision maker who is neither pessimistic nor optimistic.– It assumes all the states of nature are equally likely to
occur.– The procedure to find an optimal decision.
• For each decision add all the payoffs.• Select the decision with the largest sum (for profits).
Decision Making Under Uncertainty - Decision Making Under Uncertainty - The Principle of Insufficient ReasonThe Principle of Insufficient Reason
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TOM BROWNTOM BROWN - - Insufficient ReasonInsufficient Reason
• Sum of Payoffs– Gold 600 Dollars– Bond 350 Dollars– Stock 50 Dollars– C/D 300 Dollars
• Based on this criterion the optimal decision alternative is to invest in gold.
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Decision Making Under Uncertainty – Decision Making Under Uncertainty – Spreadsheet templateSpreadsheet template
Payoff Table
Large Rise Small Rise No Change Small Fall Large FallGold -100 100 200 300 0Bond 250 200 150 -100 -150Stock 500 250 100 -200 -600C/D Account 60 60 60 60 60d5d6d7d8Probability 0.2 0.3 0.3 0.1 0.1
Criteria Decision PayoffMaximin C/D Account 60Minimax Regret Bond 400Maximax Stock 500Insufficient Reason Gold 100EV Bond 130EVPI 141
RESULTS
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Decision Making Under RiskDecision Making Under Risk
• The probability estimate for the occurrence of
each state of nature (if available) can be
incorporated in the search for the optimal
decision.
• For each decision calculate its expected payoff.
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Decision Making Under Risk –Decision Making Under Risk –The Expected Value CriterionThe Expected Value Criterion
Expected Payoff = (Probability)(Payoff)Expected Payoff = (Probability)(Payoff)
• For each decision calculate the expected payoff as follows:
(The summation is calculated across all the states of nature)
• Select the decision with the best expected payoff
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TOM BROWN -TOM BROWN - The Expected Value CriterionThe Expected Value Criterion
The Expected Value Criterion ExpectedDecision Large rise Small rise No change Small fall Large fall ValueGold -100 100 200 300 0 100Bond 250 200 150 -100 -150 130Stock 500 250 100 -200 -600 125C/D 60 60 60 60 60 60Prior Prob. 0.2 0.3 0.3 0.1 0.1
EV = (0.2)(250) + (0.3)(200) + (0.3)(150) + (0.1)(-100) + (0.1)(-150) = 130
The optimal decision
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• The expected value criterion is useful generally in two cases:– Long run planning is appropriate, and decision
situations repeat themselves.– The decision maker is risk neutral.
When to use the expected value When to use the expected value approachapproach
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The Expected Value Criterion - The Expected Value Criterion - spreadsheetspreadsheet
=SUMPRODUCT(B4:F4,$B$8:$F$8)Drag to G7
Cell H4 (hidden) = A4Drag to H7
=MAX(G4:G7)
=VLOOKUP(MAX(G4:G7),G4:H7,2,FALSE)
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6.4 Expected Value of Perfect Information6.4 Expected Value of Perfect Information
• The gain in expected return obtained from knowing with certainty the future state of nature is called:
Expected Value of Perfect Information Expected Value of Perfect Information
(EVPI)(EVPI)
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The Expected Value of Perfect Information Decision Large rise Small rise No change Small fall Large fallGold -100 100 200 300 0Bond 250 200 150 -100 -150Stock 500 250 100 -200 -600C/D 60 60 60 60 60Probab. 0.2 0.3 0.3 0.1 0.1
If it were known with certainty that there will be a “Large Rise” in the market
Large rise
... the optimal decision would be to invest in...
-100 250
500 60
Stock
Similarly,…
TOM BROWN -TOM BROWN - EVPIEVPI
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The Expected Value of Perfect Information Decision Large rise Small rise No change Small fall Large fallGold -100 100 200 300 0Bond 250 200 150 -100 -150Stock 500 250 100 -200 -600C/D 60 60 60 60 60Probab. 0.2 0.3 0.3 0.1 0.1
-100 250
500 60
Expected Return with Perfect information = ERPI = 0.2(500)+0.3(250)+0.3(200)+0.1(300)+0.1(60) = $271
Expected Return without additional information = Expected Return of the EV criterion = $130
EVPI = ERPI - EREV = $271 - $130 = $141
TOM BROWN -TOM BROWN - EVPIEVPI
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6.5 Bayesian Analysis - Decision Making 6.5 Bayesian Analysis - Decision Making with Imperfect Informationwith Imperfect Information
• Bayesian Statistics play a role in assessing additional information obtained from various sources.
• This additional information may assist in refining original probability estimates, and help improve decision making.
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TOM BROWN – Using Sample InformationTOM BROWN – Using Sample Information
• Tom can purchase econometric forecast results for $50.
• The forecast predicts “negative” or “positive” econometric growth.
• Statistics regarding the forecast are: The Forecast When the stock market showed a... predicted Large Rise Small Rise No Change Small Fall Large Fall
Positive econ. growth 80% 70% 50% 40% 0%Negative econ. growth 20% 30% 50% 60% 100%
When the stock market showed a large rise the Forecast predicted a “positive growth” 80% of the time.
Should Tom purchase the Forecast ?
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• If the expected gain resulting from the decisions made with the forecast exceeds $50, Tom should purchase the forecast.
The expected gain =
Expected payoff with forecast – EREV• To find Expected payoff with forecast Tom should
determine what to do when: – The forecast is “positive growth”,– The forecast is “negative growth”.
TOM BROWN – SolutionTOM BROWN – SolutionUsing Sample InformationUsing Sample Information
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• Tom needs to know the following probabilities– P(Large rise | The forecast predicted “Positive”) – P(Small rise | The forecast predicted “Positive”) – P(No change | The forecast predicted “Positive ”) – P(Small fall | The forecast predicted “Positive”)– P(Large Fall | The forecast predicted “Positive”) – P(Large rise | The forecast predicted “Negative ”)– P(Small rise | The forecast predicted “Negative”)– P(No change | The forecast predicted “Negative”)– P(Small fall | The forecast predicted “Negative”)– P(Large Fall) | The forecast predicted “Negative”)
TOM BROWN – SolutionTOM BROWN – SolutionUsing Sample InformationUsing Sample Information
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• Bayes’ Theorem provides a procedure to calculate these probabilities
P(B|Ai)P(Ai)
P(B|A1)P(A1)+ P(B|A2)P(A2)+…+ P(B|An)P(An)P(Ai|B) =
Posterior ProbabilitiesProbabilities determinedafter the additional infobecomes available.
TOM BROWN – SolutionTOM BROWN – SolutionBayes’ TheoremBayes’ Theorem
Prior probabilitiesProbability estimatesdetermined based on current info, before thenew info becomes available.
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States of Prior Prob. Joint PosteriorNature Prob. (State|Positive) Prob. Prob.Large Rise 0.2 0.8 0.16 0.286Small Rise 0.3 0.7 0.21 0.375No Change 0.3 0.5 0.15 0.268Small Fall 0.1 0.4 0.04 0.071Large Fall 0.1 0 0 0.000
X =
TOM BROWN – SolutionTOM BROWN – SolutionBayes’ TheoremBayes’ Theorem
The Probability that the forecast is “positive” and the stock market shows “Large Rise”.
• The tabular approach to calculating posterior probabilities for “positive” economical forecast
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States of Prior Prob. Joint PosteriorNature Prob. (State|Positive) Prob. Prob.Large Rise 0.2 0.8 0.16 0.286Small Rise 0.3 0.7 0.21 0.375No Change 0.3 0.5 0.15 0.268Small Fall 0.1 0.4 0.04 0.071Large Fall 0.1 0 0 0.000
X =0.16 0.56
The probability that the stock market shows “Large Rise” given that the forecast is “positive”
• The tabular approach to calculating posterior probabilities for “positive” economical forecast
TOM BROWN – SolutionTOM BROWN – SolutionBayes’ TheoremBayes’ Theorem
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States of Prior Prob. Joint PosteriorNature Prob. (State|Positive) Prob. Prob.Large Rise 0.2 0.8 0.16 0.286Small Rise 0.3 0.7 0.21 0.375No Change 0.3 0.5 0.15 0.268Small Fall 0.1 0.4 0.04 0.071Large Fall 0.1 0 0 0.000
X =
TOM BROWN – SolutionTOM BROWN – SolutionBayes’ TheoremBayes’ Theorem
Observe the revision in the prior probabilities
Probability(Forecast = positive) = .56
• The tabular approach to calculating posterior probabilities for “positive” economical forecast
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States of Prior Prob. Joint PosteriorNature Prob. (State|negative) Probab. Probab.Large Rise 0.2 0.2 0.04 0.091Small Rise 0.3 0.3 0.09 0.205No Change 0.3 0.5 0.15 0.341Small Fall 0.1 0.6 0.06 0.136Large Fall 0.1 1 0.1 0.227
TOM BROWN – SolutionTOM BROWN – SolutionBayes’ TheoremBayes’ Theorem
Probability(Forecast = negative) = .44
• The tabular approach to calculating posterior probabilities for “negative” economical forecast
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Posterior (revised) ProbabilitiesPosterior (revised) Probabilitiesspreadsheet templatespreadsheet template
Bayesian Analysis
Indicator 1 Indicator 2
States Prior Conditional Joint Posterior States Prior Conditional Joint Posteriorof Nature Probabilities Probabilities Probabilities Probabilites of Nature Probabilities Probabilities Probabilities Probabilites
Large Rise 0.2 0.8 0.16 0.286 Large Rise 0.2 0.2 0.04 0.091Small Rise 0.3 0.7 0.21 0.375 Small Rise 0.3 0.3 0.09 0.205No Change 0.3 0.5 0.15 0.268 No Change 0.3 0.5 0.15 0.341Small Fall 0.1 0.4 0.04 0.071 Small Fall 0.1 0.6 0.06 0.136Large Fall 0.1 0 0 0.000 Large Fall 0.1 1 0.1 0.227s6 0 0 0.000 s6 0 0 0.000s7 0 0 0.000 s7 0 0 0.000s8 0 0 0.000 s8 0 0 0.000
P(Indicator 1) 0.56 P(Indicator 2) 0.44
Bayesian Analysis
Indicator 1 Indicator 2
States Prior Conditional Joint Posterior States Prior Conditional Joint Posteriorof Nature Probabilities Probabilities Probabilities Probabilites of Nature Probabilities Probabilities Probabilities Probabilites
Large Rise 0.2 0.8 0.16 0.286 Large Rise 0.2 0.2 0.04 0.091Small Rise 0.3 0.7 0.21 0.375 Small Rise 0.3 0.3 0.09 0.205No Change 0.3 0.5 0.15 0.268 No Change 0.3 0.5 0.15 0.341Small Fall 0.1 0.4 0.04 0.071 Small Fall 0.1 0.6 0.06 0.136Large Fall 0.1 0 0 0.000 Large Fall 0.1 1 0.1 0.227s6 0 0 0.000 s6 0 0 0.000s7 0 0 0.000 s7 0 0 0.000s8 0 0 0.000 s8 0 0 0.000
P(Indicator 1) 0.56 P(Indicator 2) 0.44
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• This is the expected gain from making decisions based on Sample Information.
• Revise the expected return for each decision using the posterior probabilities as follows:
Expected Value of Sample Expected Value of Sample InformationInformation
EVSIEVSI
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The revised probabilities payoff tableDecision Large rise Small rise No change Small fall Large fall
Gold -100 100 200 300 0
Bond 250 200 150 -100 -150
Stock 500 250 100 -200 -600
C/D 60 60 60 60 60P(State|Positive) 0.286 0.375 0.268 0.071 0
P(State|negative) 0.091 0.205 0.341 0.136 0.227
EV(Invest in……. |“Positive” forecast) = =.286( )+.375( )+.268( )+.071( )+0( ) =
EV(Invest in ……. | “Negative” forecast) =
=.091( )+.205( )+.341( )+.136( )+.227( ) =
-100 100 200 300 $840GOLD
-100 100 200 300 0
GOLD
$120
TOM BROWN – Conditional Expected ValuesTOM BROWN – Conditional Expected Values
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• The revised expected values for each decision:Positive forecast Negative forecastEV(Gold|Positive) = 84 EV(Gold|Negative) = 120EV(Bond|Positive) = 180 EV(Bond|Negative) = 65EV(Stock|Positive) = 250 EV(Stock|Negative) = -37 EV(C/D|Positive) = 60 EV(C/D|Negative) = 60
If the forecast is “Positive”Invest in Stock.
If the forecast is “Negative”Invest in Gold.
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• Since the forecast is unknown before it is purchased, Tom can only calculate the expected return from purchasing it.
• Expected return when buying the forecast = ERSI = P(Forecast is positive)(EV(Stock|Forecast is positive)) + P(Forecast is negative”)(EV(Gold|Forecast is negative)) = (.56)(250) + (.44)(120) = $192.5
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• The expected gain from buying the forecast is:EVSI = ERSI – EREV = 192.5 – 130 = $62.5
• Tom should purchase the forecast. His expected gain is greater than the forecast cost.
• Efficiency = EVSI / EVPI = 63 / 141 = 0.45
Expected Value of Sampling Expected Value of Sampling Information (EVSI)Information (EVSI)
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TOM BROWN – SolutionTOM BROWN – SolutionEVSI spreadsheet templateEVSI spreadsheet template
Payoff Table
Large Rise Small Rise No Change Small Fall Large Fall s6 s7 s8 EV(prior) EV(ind. 1) EV(ind. 2)Gold -100 100 200 300 0 100 83.93 120.45Bond 250 200 150 -100 -150 130 179.46 67.05Stock 500 250 100 -200 -600 125 249.11 -32.95C/D Account 60 60 60 60 60 60 60.00 60.00d5d6d7d8Prior Prob. 0.2 0.3 0.3 0.1 0.1Ind. 1 Prob. 0.286 0.375 0.268 0.071 0.000 #### ### ## 0.56Ind 2. Prob. 0.091 0.205 0.341 0.136 0.227 #### ### ## 0.44Ind. 3 Prob.Ind 4 Prob.
RESULTSPrior Ind. 1 Ind. 2 Ind. 3 Ind. 4
optimal payoff 130.00 249.11 120.45 0.00 0.00optimal decision Bond Stock Gold
EVSI = 62.5EVPI = 141Efficiency= 0.44
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6.6 Decision Trees6.6 Decision Trees
• The Payoff Table approach is useful for a non-sequential or single stage.
• Many real-world decision problems consists of a sequence of dependent decisions.
• Decision Trees are useful in analyzing multi-stage decision processes.
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• A Decision Tree is a chronological representation of the decision process.
• The tree is composed of nodes and branches.
Characteristics of a decision treeCharacteristics of a decision tree
A branch emanating from a state of nature (chance) node corresponds to a particular state of nature, and includes the probability of this state of nature.
Decision node
Chance node
Decision 1
Cost 1Decision 2Cost 2
P(S2)
P(S 1)
P(S3 )
P(S2)
P(S 1)
P(S3 )
A branch emanating from a decision node corresponds to a decision alternative. It includes a cost or benefit value.
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BILL GALLEN DEVELOPMENT COMPANYBILL GALLEN DEVELOPMENT COMPANY
– BGD plans to do a commercial development on a property.
– Relevant data• Asking price for the property is 300,000 dollars.• Construction cost is 500,000 dollars.• Selling price is approximated at 950,000 dollars.• Variance application costs 30,000 dollars in fees and expenses
– There is only 40% chance that the variance will be approved.– If BGD purchases the property and the variance is denied, the property
can be sold for a net return of 260,000 dollars.– A three month option on the property costs 20,000 dollars, which will
allow BGD to apply for the variance.
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– A consultant can be hired for 5000 dollars.– The consultant will provide an opinion about the
approval of the application • P (Consultant predicts approval | approval granted) = 0.70• P (Consultant predicts denial | approval denied) = 0.80
• BGD wishes to determine the optimal strategy– Hire/ not hire the consultant now,– Other decisions that follow sequentially.
BILL GALLEN DEVELOPMENT COMPANYBILL GALLEN DEVELOPMENT COMPANY
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BILL GALLEN - SolutionBILL GALLEN - Solution
• Construction of the Decision Tree – Initially the company faces a decision about hiring the
consultant.
– After this decision is made more decisions follow regarding • Application for the variance.• Purchasing the option.• Purchasing the property.
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BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree
Let us consider the decision
to not hire a consultant
Do not hire consultant
Hire consultantCost = -5000
Cost = 0
Do nothing
0Buy land-300,000Purchase option
-20,000
Apply for variance
Apply for variance
-30,000
-30,000
03
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Approved
Denied
0.4
0.6
12
Approved
Denied
0.4
0.6
-300,000 -500,000 950,000
Buy land Build Sell
-50,000
100,000
-70,000
260,000Sell
Build Sell950,000-500,000
120,000Buy land and apply for variance
-300000 – 30000 + 260000 =
-300000 – 30000 – 500000 + 950000 =
Purchase option andapply for variance
BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree
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60
Do not hire consultant
Hire consultantCost = -5000
Cost = 0
Do nothing
0
Buy land-300,000Purchase option
-20,000
Apply for variance
Apply for variance
-30,000
-30,000
0
61
12
-300,000 -500,000 950,000
Buy land Build Sell
-50,000
100,000
-70,000
260,000Sell
Build Sell950,000-500,000
120,000Buy land and apply for variance
-300000 – 30000 + 260000 =
-300000 – 30000 – 500000 + 950000 =
Purchase option andapply for variance
This is where we are at this stage
Let us consider the decision to hire a consultant
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Do not hire consultant
0
Hire consultant-5000 Predic
t
Approval
Predict
Denial
0.4
0.6
-5000
Apply for variance
Apply for variance
Apply for variance
Apply for variance
-5000
-30,000
-30,000
-30,000
-30,000
BILL GALLEN – BILL GALLEN – The Decision Tree The Decision Tree
Let us consider the decision to hire a consultant
Done
Do Nothing
Buy land-300,000
Purchase option-20,000
Do Nothing
Buy land-300,000
Purchase option-20,000
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BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree
Approved
Denied
Consultant predicts an approval
?
?
Build Sell950,000-500,000
260,000Sell
-75,000
115,000
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BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree
Approved
Denied?
?
Build Sell950,000-500,000
260,000Sell
-75,000
115,000
The consultant serves as a source for additional information about denial or approval of the variance.
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?
?
BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree
Approved
Denied
Build Sell950,000-500,000
260,000Sell
-75,000
115,000
Therefore, at this point we need to calculate theposterior probabilities for the approval and denial
of the variance application
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BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree
22
Approved
Denied
Build Sell950,000-500,000
260,000Sell
-75,000
27
25115,000
23 24
26
The rest of the Decision Tree is built in a similar manner.
Posterior Probability of (approval | consultant predicts approval) = 0.70Posterior Probability of (denial | consultant predicts approval) = 0.30
?
?
.7
.3
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• Work backward from the end of each branch.
• At a state of nature node, calculate the expected value of the node.
• At a decision node, the branch that has the highest ending node value represents the optimal decision.
The Decision TreeThe Decision Tree Determining the Optimal Strategy Determining the Optimal Strategy
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22
Approved
Denied
27
2523 24
26-75,000
115,000115,000
-75,000
115,000
-75,000
115,000
-75,000
115,000
-75,00022
115,000
-75,000
(115,000)(0.7)=80500
(-75,000)(0.3)= -22500
-22500
80500
80500
-22500
80500
-22500
58,000?
?0.30
0.70
Build Sell950,000-500,000
260,000Sell
-75,000
115,000
With 58,000 as the chance node value,we continue backward to evaluate
the previous nodes.
BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree Determining the Optimal Strategy Determining the Optimal Strategy
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Predicts approvalHire
Do nothing
BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree Determining the Optimal Strategy Determining the Optimal Strategy
.4
.6
$10,000
$58,000
$-5,000
$20,000
$20,000
Buy land; Apply for variance
Predicts denial
Denied
Build,Sell
Sell land
Do not
hire
$-75,000
$115,000
.7
.3
Appr
oved
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BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree Excel add-in: Tree Plan Excel add-in: Tree Plan
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BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree Excel add-in: Tree Plan Excel add-in: Tree Plan
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6.7 Decision Making and Utility6.7 Decision Making and Utility
• Introduction– The expected value criterion may not be appropriate
if the decision is a one-time opportunity with substantial risks.
– Decision makers do not always choose decisions based on the expected value criterion.
• A lottery ticket has a negative net expected return.• Insurance policies cost more than the present value of the
expected loss the insurance company pays to cover insured losses.
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• It is assumed that a decision maker can rank decisions in a coherent manner.
• Utility values, U(V), reflect the decision maker’s perspective and attitude toward risk.
• Each payoff is assigned a utility value. Higher payoffs get larger utility value.
• The optimal decision is the one that maximizes the expected utility.
The Utility ApproachThe Utility Approach
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• The technique provides an insightful look into the amount of risk the decision maker is willing to take.
• The concept is based on the decision maker’s preference to taking a sure payoff versus participating in a lottery.
Determining Utility ValuesDetermining Utility Values
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• List every possible payoff in the payoff table in ascending order.
• Assign a utility of 0 to the lowest value and a value of 1 to the highest value.
• For all other possible payoffs (Rij) ask the decision maker the following question:
Determining Utility ValuesDetermining Utility Values Indifference approach for assigning utility valuesIndifference approach for assigning utility values
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• Suppose you are given the option to select one of the following two alternatives:– Receive $Rij (one of the payoff values) for sure, – Play a game of chance where you receive either
• The highest payoff of $Rmax with probability p, or
• The lowest payoff of $Rmin with probability 1- p.
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Rmin
What value of p would make you indifferent between the two situations?”
Determining Utility ValuesDetermining Utility Values Indifference approach for assigning utility valuesIndifference approach for assigning utility values
Rij
Rmax
p
1-p
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Rmin
The answer to this question is the indifference probability for the payoff Rij and is used as the utility values of Rij.
Determining Utility ValuesDetermining Utility Values Indifference approach for assigning utility valuesIndifference approach for assigning utility values
Rij
Rmax
p
1-p
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Determining Utility ValuesDetermining Utility Values Indifference approach for assigning utility valuesIndifference approach for assigning utility values
d1
d2
s1 s1
150
-50 140
100
Alternative 1A sure event
Alternative 2 (Game-of-chance)
$100$150
-50p1-p
• For p = 1.0, you’ll prefer Alternative 2.• For p = 0.0, you’ll prefer Alternative 1.• Thus, for some p between 0.0 and 1.0 you’ll be indifferent between the alternatives.
Example:
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Determining Utility ValuesDetermining Utility Values Indifference approach for assigning utility valuesIndifference approach for assigning utility values
d1
d2
s1 s1
150
-50 140
100
Alternative 1A sure event
Alternative 2 (Game-of-chance)
$100$150
-50p1-p
• Let’s assume the probability of indifference is p = .7. U(100)=.7U(150)+.3U(-50) = .7(1) + .3(0) = .7
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TOM BROWNTOM BROWN - - Determining Utility ValuesDetermining Utility Values• Data
– The highest payoff was $500. Lowest payoff was -$600.– The indifference probabilities provided by Tom are
– Tom wishes to determine his optimal investment Decision.
Payoff -600 -200 -150 -100 0 60 100 150 200 250 300 500
Prob. 0 0.25 0.3 0.36 0.5 0.6 0.65 0.7 0.75 0.85 0.9 1
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TOM BROWNTOM BROWN – – Optimal decision (utility)Optimal decision (utility)
Utility Analysis Certain Payoff Utility-600 0
Large Rise Small Rise No Change Small Fall Large Fall EU -200 0.25Gold 0.36 0.65 0.75 0.9 0.5 0.632 -150 0.3Bond 0.85 0.75 0.7 0.36 0.3 0.671 -100 0.36Stock 1 0.85 0.65 0.25 0 0.675 0 0.5C/D Account 0.6 0.6 0.6 0.6 0.6 0.6 60 0.6d5 0 100 0.65d6 0 150 0.7d7 0 200 0.75d8 0 250 0.85Probability 0.2 0.3 0.3 0.1 0.1 300 0.9
500 1
RESULTSCriteria Decision ValueExp. Utility Stock 0.675
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Three types of Decision MakersThree types of Decision Makers
• Risk Averse -Prefers a certain outcome to a chance outcome having the same expected value.
• Risk Taking - Prefers a chance outcome to a certain outcome having the same expected value.
• Risk Neutral - Is indifferent between a chance outcome and a certain outcome having the same expected value.
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80Payoff
UtilityThe Utility Curve for a Risk Averse Decision Maker
1000.5
2000.5
150
The utility of having $150 on hand…The utility of having $150 on hand…
U(150)
…is larger than the expected utilityof a game whose expected valueis also $150.
…is larger than the expected utilityof a game whose expected valueis also $150.
EU(Game)
U(100)
U(200)
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81Payoff
Utility
1000.5
2000.5
150
U(150)EU(Game)
U(100)
U(200)
A risk averse decision maker avoidsthe thrill of a game-of-chance,whose expected value is EV, if he can have EV on hand for sure.
A risk averse decision maker avoidsthe thrill of a game-of-chance,whose expected value is EV, if he can have EV on hand for sure.
CE
Furthermore, a risk averse decision maker is willing to pay a premium…
Furthermore, a risk averse decision maker is willing to pay a premium…
…to buy himself (herself) out of the game-of-chance.
…to buy himself (herself) out of the game-of-chance.
The Utility Curve for a Risk Averse Decision Maker
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Risk Neutral D
ecision Maker
Payoff
UtilityRisk Averse Decision Maker
Risk Taking Decision Maker