Ch06Decision Tree inductive learning task.ppt

A Decision Tree is presented here as an inductive learning task, meaning it learns from specific examples or data to make generalized conclusions. In other words, it observes patterns from known cases (facts) and uses them to predict outcomes for new, unseen situations.

1. 1
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.

2. 2
•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

3. 3
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.

4. 4
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

5. 5
•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

6. 6
S1S2S3S4
D1p
11
p
12
p
13
p
14
D2p
21p
22p
23P
24
D3p
31p
32p
33p
34
•Select a decision making criterion, and
apply it to the payoff table.
TOM BROWN - SolutionTOM BROWN - Solution
S1S2S3S4
D1p
11
p
12
p
13
p
14
D2p
21p
22p
23P
24
D3p
31p
32p
33p
34
Criterion
P1
P2
P3
•Construct a payoff table.
•Identify the optimal decision.
•Evaluate the solution.

7. 7
Decision States of Nature
AlternativesLarge RiseSmall RiseNo ChangeSmall FallLarge Fall
Gold -100 100 200 300 0
Bond 250 200 150 -100 -150
Stock 500 250 100 -200 -600
C/D account 60 60 60 60 60
Stock option200 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 moves
within
[-300,+300]
DJA is up
[+300,+1000]
DJA is up more
than1000 points

8. 8
Decision States of Nature
AlternativesLarge RiseSmall RiseNo ChangeSmall FallLarge Fall
Gold -100 100 200 300 0
Bond 250 200 150 -100 -150
Stock 500 250 100 -200 -600
C/D account 60 60 60 60 60
Stock option200 150 150 -200 -150
The Payoff TableThe Payoff Table
Determine the
set of possible
decision
alternatives.

9. 9
Decision States of Nature
AlternativesLarge RiseSmall RiseNo ChangeSmall FallLarge Fall
Gold -100 100 200 300 0
Bond 250 200 150 -100 -150
Stock 500 250 100 -200 -600
C/D account 60 60 60 60 60
Stock option200 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

10. 10
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.

11. 11
•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

12. 12
Decision Making Under Uncertainty - Decision Making Under Uncertainty -
The Maximin CriterionThe Maximin Criterion

13. 13
•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

14. 14
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 FallPayoff
Gold -100 100 200 300 0 -100
Bond 250 200 150 -100-150-150
Stock 500 250 100 -200-600-600
C/D account 60 60 60 60 60 60
The Maximin Criterion Minimum
Decisions Large Rise Small rise No Change Small Fall Large FallPayoff
Gold -100 100 200 300 0 -100
Bond 250 200 150 -100-150-150
Stock 500 250 100 -200-600-600
C/D account 60 60 60 60 60 60
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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

16. 16
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)=A4
Drag to I7
The Maximin Criterion - spreadsheetThe Maximin Criterion - spreadsheet

17. 17
Decision Making Under Uncertainty - Decision Making Under Uncertainty -
The Minimax Regret CriterionThe Minimax Regret Criterion

18. 18
• 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

19. 19
•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

20. 20
•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

21. 21
•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

22. 22
TOM BROWN -TOM BROWN - The Maximax CriterionThe Maximax Criterion
The Maximax Criterion Maximum
DecisionLarge riseSmall riseNo changeSmall fallLarge fallPayoff
Gold -100 100 200 300 0 300
Bond 250 200 150 -100-150 200
Stock 500 250 100 -200-600 500
C/D 60 60 60 60 60 60
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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

24. 24
TOM BROWNTOM BROWN - - Insufficient ReasonInsufficient Reason
• Sum of Payoffs
–Gold600 Dollars
–Bond350 Dollars
–Stock 50 Dollars
–C/D300 Dollars
•Based on this criterion the optimal decision
alternative is to invest in gold.

25. 25
Decision Making Under Uncertainty – Decision Making Under Uncertainty –
Spreadsheet templateSpreadsheet template
Payoff Table
Large RiseSmall RiseNo ChangeSmall FallLarge Fall
Gold -100 100 200 300 0
Bond 250 200 150 -100 -150
Stock 500 250 100 -200 -600
C/D Account 60 60 60 60 60
d5
d6
d7
d8
Probability 0.2 0.3 0.3 0.1 0.1
Criteria Decision Payoff
Maximin C/D Account 60
Minimax Regret Bond 400
Maximax Stock 500
Insufficient ReasonGold 100
EV Bond 130
EVPI 141
RESULTS

26. 26
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.

27. 27
Decision Making Under Risk –Decision Making Under Risk –
The Expected Value CriterionThe Expected Value Criterion
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

28. 28
TOM BROWN -TOM BROWN - The Expected Value CriterionThe Expected Value Criterion
The Expected Value Criterion Expected
DecisionLarge riseSmall riseNo changeSmall fallLarge fallValue
Gold -100 100 200 300 0 100
Bond 250 200 150 -100-150 130
Stock 500 250 100 -200-600 125
C/D 60 60 60 60 60 60
Prior 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
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29. 29
•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

30. 30
The Expected Value Criterion - The Expected Value Criterion -
spreadsheetspreadsheet
=SUMPRODUCT(B4:F4,$B$8:$F$8)
Drag to G7
Cell H4 (hidden) = A4
Drag to H7
=MAX(G4:G7)
=VLOOKUP(MAX(G4:G7),G4:H7,2,FALSE)

31. 31
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)

32. 32
The Expected Value of Perfect Information
DecisionLarge riseSmall riseNo changeSmall fallLarge 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 60
Probab. 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

33. 33
The Expected Value of Perfect Information
DecisionLarge riseSmall riseNo changeSmall fallLarge 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 60
Probab. 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

34. 34
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.

35. 35
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 RiseSmall RiseNo ChangeSmall FallLarge Fall
Positive econ. growth80% 70% 50% 40% 0%
Negative econ. growth20% 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 ?

36. 36
•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 – Solution
Using Sample InformationUsing Sample Information

37. 37
•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 – Solution
Using Sample InformationUsing Sample Information

38. 38
•Bayes’ Theorem provides a procedure to calculate
these probabilities
P(B|A
i
)P(A
i
)
P(B|A
1
)P(A
1
)+ P(B|A
2
)P(A
2
)+…+ P(B|A
n
)P(A
n
)
P(A
i|B) =
Posterior Probabilities
Probabilities determined
after the additional info
becomes available.
TOM BROWN – SolutionTOM BROWN – Solution
Bayes’ TheoremBayes’ Theorem
Prior probabilities
Probability estimates
determined based on
current info, before the
new info becomes available.

39. 39
States of Prior Prob. Joint Posterior
Nature Prob.(State|Positive)Prob. Prob.
Large Rise 0.2 0.8 0.16 0.286
Small Rise 0.3 0.7 0.21 0.375
No Change 0.3 0.5 0.15 0.268
Small Fall 0.1 0.4 0.04 0.071
Large Fall 0.1 0 0 0.000
X =
TOM BROWN – SolutionTOM BROWN – Solution
Bayes’ 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

40. 40
States of Prior Prob. Joint Posterior
Nature Prob.(State|Positive)Prob. Prob.
Large Rise 0.2 0.8 0.16 0.286
Small Rise 0.3 0.7 0.21 0.375
No Change 0.3 0.5 0.15 0.268
Small Fall 0.1 0.4 0.04 0.071
Large 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 – Solution
Bayes’ TheoremBayes’ Theorem

41. 41
States of Prior Prob. Joint Posterior
Nature Prob.(State|Positive)Prob. Prob.
Large Rise 0.2 0.8 0.16 0.286
Small Rise 0.3 0.7 0.21 0.375
No Change 0.3 0.5 0.15 0.268
Small Fall 0.1 0.4 0.04 0.071
Large Fall 0.1 0 0 0.000
X =
TOM BROWN – SolutionTOM BROWN – Solution
Bayes’ TheoremBayes’ Theorem
Observe the revision in
the prior probabilities
Probability(Forecast = positive) = .56
•The tabular approach to calculating posterior
probabilities for “positive” economical forecast

42. 42
States of Prior Prob. Joint Posterior
Nature Prob.(State|negative)Probab.Probab.
Large Rise 0.2 0.2 0.04 0.091
Small Rise 0.3 0.3 0.09 0.205
No Change 0.3 0.5 0.15 0.341
Small Fall 0.1 0.6 0.06 0.136
Large Fall 0.1 1 0.1 0.227
TOM BROWN – SolutionTOM BROWN – Solution
Bayes’ TheoremBayes’ Theorem
Probability(Forecast = negative) = .44
•The tabular approach to calculating posterior
probabilities for “negative” economical forecast

43. 43
Posterior (revised) ProbabilitiesPosterior (revised) Probabilities
spreadsheet templatespreadsheet template
Bayesian Analysis
Indicator 1 Indicator 2
States Prior ConditionalJoint PosteriorStates Prior ConditionalJoint Posterior
of NatureProbabilitiesProbabilitiesProbabilitiesProbabilitesof NatureProbabilitiesProbabilitiesProbabilitiesProbabilites
Large Rise 0.2 0.8 0.16 0.286 Large Rise 0.2 0.2 0.04 0.091
Small Rise 0.3 0.7 0.21 0.375 Small Rise 0.3 0.3 0.09 0.205
No Change 0.3 0.5 0.15 0.268 No Change 0.3 0.5 0.15 0.341
Small Fall 0.1 0.4 0.04 0.071 Small Fall 0.1 0.6 0.06 0.136
Large Fall 0.1 0 0 0.000 Large Fall 0.1 1 0.1 0.227
s6 0 0 0.000 s6 0 0 0.000
s7 0 0 0.000 s7 0 0 0.000
s8 0 0 0.000 s8 0 0 0.000
P(Indicator 1)0.56 P(Indicator 2)0.44

44. 44
•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

45. 45
The revised probabilities payoff table
Decision Large riseSmall riseNo changeSmall fallLarge fall
Gold -100100 200300 0
Bond 250 200 150-100 -150
Stock 500 250 100-200 -600
C/D 60 60 60 60 60
P(State|Positive)0.2860.3750.2680.071 0
P(State|negative)0.0910.2050.3410.1360.227
EV(Invest in……. |“Positive” forecast) =
=.286( )+.375( )+.268( )+.071( )+0( ) =
EV(Invest in ……. | “Negative” forecast) =
=.091( )+.205( )+.341( )+.136( )+.227( ) =
-100 100 200 300 $840
GOLD
-100 100 200 300 0
GOLD
$120
TOM BROWN – Conditional Expected ValuesTOM BROWN – Conditional Expected Values

46. 46
•The revised expected values for each decision:
Positive forecastNegative forecast
EV(Gold|Positive) = 84EV(Gold|Negative) = 120
EV(Bond|Positive) = 180EV(Bond|Negative) = 65
EV(Stock|Positive) = 250 EV(Stock|Negative)
= -37
EV(C/D|Positive) = 60EV(C/D|Negative) = 60
If the forecast is “Positive”
Invest in Stock.
If the forecast is “Negative”
Invest in Gold.
TOM BROWN – Conditional Expected ValuesTOM BROWN – Conditional Expected Values

47. 47
•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
TOM BROWN – Conditional Expected ValuesTOM BROWN – Conditional Expected Values

48. 48
•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)

49. 49
TOM BROWN – SolutionTOM BROWN – Solution
EVSI spreadsheet templateEVSI spreadsheet template
Payoff Table
Large RiseSmall RiseNo ChangeSmall FallLarge Falls6s7s8EV(prior)EV(ind. 1)EV(ind. 2)
Gold -100 100 200 300 0 100 83.93 120.45
Bond 250 200 150 -100 -150 130 179.46 67.05
Stock 500 250 100 -200 -600 125 249.11 -32.95
C/D Account 60 60 60 60 60 60 60.00 60.00
d5
d6
d7
d8
Prior Prob. 0.2 0.3 0.3 0.1 0.1
Ind. 1 Prob. 0.286 0.375 0.268 0.071 0.000######### 0.56
Ind 2. Prob. 0.091 0.205 0.341 0.136 0.227######### 0.44
Ind. 3 Prob.
Ind 4 Prob.
RESULTS
Prior Ind. 1 Ind. 2 Ind. 3Ind. 4
optimal payoff 130.00 249.11 120.45 0.00 0.00
optimal decision Bond Stock Gold
EVSI = 62.5
EVPI = 141
Efficiency= 0.44

50. 50
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.

51. 51
•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 1
D
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2C
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P(S
2
)
P(S1
)
P
(
S
3)
P(S
2)
P(S1
)
P
(
S
3)
A branch emanating from a
decision node corresponds to a
decision alternative. It includes a
cost or benefit value.

52. 52
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.

53. 53
–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

54. 54
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.

55. 55
BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree
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=

-
5
0
0
0
C
ost = 0
Do nothing
0
Buy land
-300,000
P
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-
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Apply for variance
Apply for variance
-30,000
-30,000
0
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Approved
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6
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Approved
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-300,000 -500,000 950,000
Buy land Build Sell
-50,000
100,000
-70,000
260,000
Sell
Build Sell
950,000-500,000
120,000Buy land and
apply for variance
-300000 – 30000 + 260000 =
-300000 – 30000 – 500000 + 950000 =
Purchase option and
apply for variance
BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree

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D
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-
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Do nothing
0
Buy land
-300,000
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-
2
0
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0
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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
260,000
Sell
Build Sell
950,000-500,000
120,000Buy land and
apply for variance
-300000 –30000 + 260000 =
-300000 –30000 –500000 + 950000 =
Purchase option and
apply for variance
This is where we are at this stage
Let us consider the decision to hire a consultant
BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree

58. 58
Do not hire consultant
0
H
i
r
e

c
o
n
s
u
l
t
a
n
t
-
5
0
0
0 P
re d ic t
A
p p ro v a l
P
r
e
d
i
c
t
D
e
n
i
a
l
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
P
u
rch
a
se
o
p
tio
n-2
0
,0
0
0
Do Nothing
Buy land
-300,000
P
u
rc
h
a
s
e
o
p
tio
n
-2
0
,0
0
0

59. 59
BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree
Approved
D
e
n
ie
d
C
o
n
s
u
l
t
a
n
t

p
r
e
d
i
c
t
s

a
n

a
p
p
r
o
v
a
l
?
?
Build Sell
950,000-500,000
260,000
Sell
-75,000
115,000

60. 60
BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree
Approved
D
e
n
ie
d
?
?
Build Sell
950,000-500,000
260,000
Sell
-75,000
115,000
The consultant serves as a source for additional information
about denial or approval of the variance.

61. 61
?
?
BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree
Approved
D
e
n
ie
d
Build Sell
950,000-500,000
260,000
Sell
-75,000
115,000
Therefore, at this point we need to calculate the
posterior probabilities for the approval and denial
of the variance application

62. 62
BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree
22
Approved
D
e
n
ie
d
Build Sell
950,000-500,000
260,000
Sell
-75,000
27
25
115,000
23 24
26
The rest of the Decision Tree is built in a similar manner.
Posterior Probability of (approval | consultant predicts approval) = 0.70
Posterior Probability of (denial | consultant predicts approval) = 0.30
?
?
.7
.3

63. 63
•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

64. 64
22
Approved
D
e
n
ie
d
27
2523 24
26
-75,000
115,000
115,000
-75,000
115,000
-75,000
115,000
-75,000
115,000
-75,00022
115,000
-75,000
(115,000)(0.7)=80500
(
-
7
5
,
0
0
0
)
(
0
.
3
)
=

-
2
2
5
0
0
-
2
2
5
0
0
80500
80500
-
2
2
5
0
0
80500
-
2
2
5
0
0
58,000
?
?
0.30
0.70
Build Sell
950,000-500,000
260,000
Sell
-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

65. 65
Predicts approval
H
i
r
e
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
P
r
e
d
ic
ts
d
e
n
ia
l
D
e
n
i
e
d
Build,
Sell
Sell
land
D
o n o t
h ire
$-75,000
$115,000
.7
.3
A
p
p
ro
v
e
d

66. 66
BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree
Excel add-in: Tree Plan Excel add-in: Tree Plan

67. 67
BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree
Excel add-in: Tree Plan Excel add-in: Tree Plan

68. 68
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.

69. 69
•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

70. 70
•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

71. 71
• 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 (R
ij
) ask the decision
maker the following question:
Determining Utility ValuesDetermining Utility Values
Indifference approach for assigning utility valuesIndifference approach for assigning utility values

72. 72
•Suppose you are given the option to select one
of the following two alternatives:
–Receive $R
ij
(one of the payoff values) for sure,
–Play a game of chance where you receive either
•The highest payoff of $R
max
with probability p, or
•The lowest payoff of $R
min
with probability 1- p.
Determining Utility ValuesDetermining Utility Values
Indifference approach for assigning utility valuesIndifference approach for assigning utility values

73. 73
R
min
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
R
ij
R
max
p
1-p

74. 74
R
min
The answer to this question is the indifference probability
for the payoff R
ij and is used as the utility values of R
ij.
Determining Utility ValuesDetermining Utility Values
Indifference approach for assigning utility valuesIndifference approach for assigning utility values
R
ij
R
max
p
1-p

75. 75
Determining Utility ValuesDetermining Utility Values
Indifference approach for assigning utility valuesIndifference approach for assigning utility values
d
1
d
2
s
1 s
1
150
-50 140
100
Alternative 1
A sure event
Alternative 2
(Game-of-chance)
$100
$150
-50p
1-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:

76. 76
Determining Utility ValuesDetermining Utility Values
Indifference approach for assigning utility valuesIndifference approach for assigning utility values
d
1
d
2
s
1 s
1
150
-50 140
100
Alternative 1
A sure event
Alternative 2
(Game-of-chance)
$100
$150
-50p
1-p
• Let’s assume the
probability of
indifference is p = .7.

U(100)=.7U(150)+.3U(-50)
= .7(1) + .3(0) = .7

77. 77
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-600-200-150-100060100150200250300500
Prob. 00.250.30.360.50.60.650.70.750.850.91

78. 78
TOM BROWNTOM BROWN – – Optimal decision (utility)Optimal decision (utility)
Utility Analysis Certain PayoffUtility
-600 0
Large RiseSmall RiseNo ChangeSmall FallLarge FallEU -200 0.25
Gold 0.36 0.65 0.75 0.9 0.5 0.632 -150 0.3
Bond 0.85 0.75 0.7 0.36 0.3 0.671 -100 0.36
Stock 1 0.85 0.65 0.25 0 0.675 0 0.5
C/D Account 0.6 0.6 0.6 0.6 0.6 0.6 60 0.6
d5 0 100 0.65
d6 0 150 0.7
d7 0 200 0.75
d8 0 250 0.85
Probability 0.2 0.3 0.3 0.1 0.1 300 0.9
500 1
RESULTS
Criteria Decision Value
Exp. Utility Stock 0.675

79. 79
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.

80. 80
Payoff
Utility
The Utility Curve for a
Risk Averse Decision Maker
100
0.5
200
0.5
150
The utility of having $150 on hand…
U(150)
…is larger than the expected utility
of a game whose expected value
is also $150.
EU(Game)
U(100)
U(200)

81. 81
Payoff
Utility
100
0.5
200
0.5
150
U(150)
EU(Game)
U(100)
U(200)
A risk averse decision maker avoids
the 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…
…to buy himself (herself) out of the
game-of-chance.
The Utility Curve for a
Risk Averse Decision Maker

82. 82
R
isk N
eutral D
ecision M
aker
Payoff
Utility
Risk Averse Decision Maker
Risk Taking Decision Maker