Qntmeth9 Ppt Ch03
22
1 3-1 Chapter 3 Chapter 3 Decision Analysis Decision Analysis 3-2 Learning Objectives Learning Objectives Students will be able to: 1. List the steps of the decision-making process. 2. Describe the types of decision-making environments. 3. Make decisions under uncertainty. 4. Use probability values to make decisions under risk. 5. Develop accurate and useful decision trees. 6. Revise probabilities using Bayesian analysis. 7. Use computers to solve basic decision- making problems. 8. Understand the importance and use of utility theory in decision theory. 3-3 Chapter Outline Chapter Outline 3.1 Introduction 3.2 The Six Steps in Decision Theory 3.3 Types of Decision-Making Environments 3.4 Decision Making under Uncertainty 3.5 Decision Making under Risk 3.6 Decision Trees 3.7 How Probability Values Are Estimated by Bayesian Analysis 3.8 Utility Theory 3-4 Introduction Introduction Decision theory is an analytical and systematic way to tackle problems. A good decision is based on logic.
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Transcript of Qntmeth9 Ppt Ch03
1
3-1
Chapter 3Chapter 3
Decision AnalysisDecision Analysis
3-2
Learning ObjectivesLearning ObjectivesStudents will be able to:
1. List the steps of the decision-making process.
2. Describe the types of decision-making environments.
3. Make decisions under uncertainty.
4. Use probability values to make decisions under risk.
5. Develop accurate and useful decision trees.
6. Revise probabilities using Bayesian analysis.
7. Use computers to solve basic decision-making problems.
8. Understand the importance and use of utility theory in decision theory.
3-3
Chapter OutlineChapter Outline3.1 Introduction
3.2 The Six Steps in Decision Theory
3.3 Types of Decision-Making Environments
3.4 Decision Making under Uncertainty
3.5 Decision Making under Risk
3.6 Decision Trees
3.7 How Probability Values Are Estimated by Bayesian Analysis
3.8 Utility Theory
3-4
IntroductionIntroduction
� Decision theory is an analytical and systematic way to tackle problems.
� A good decision is based on logic.
2
3-5
The Six Steps in The Six Steps in Decision TheoryDecision Theory
1. Clearly define the problem at hand.
2. List the possible alternatives.
3. Identify the possible outcomes.
4. List the payoff or profit of each combination of alternatives and outcomes.
5. Select one of the mathematical decision theory models.
6. Apply the model and make your decision.
3-6
John Thompson’s John Thompson’s Backyard Storage Backyard Storage
ShedsSheds
Solutions can be obtained and a sensitivity analysis used to make a decision
Apply model and make decision
Decision tables and/or trees can be used to solve the problem
Select a model
List the payoff for each state of nature/decision alternative combination
List payoffs
The market could be favorable or unfavorable for storage sheds
Identify outcomes
1. Construct a large new plant
2. A small plant
3. No plant at all
List alternatives
To manufacture or market backyard storage sheds
Define problem
3-7
Decision Table Decision Table for Thompson Lumberfor Thompson Lumber
0
-20,000
-180,000
Unfavorable Market ($)
0
100,000
200,000
Favorable Market ($)
State of Nature
Do nothing
Construct a small plant
Construct a large plant
Alternative
3-8
Types of DecisionTypes of Decision--Making EnvironmentsMaking Environments
� Type 1: Decision making under certainty.�Decision makerknows with certainty
the consequences of every alternative or decision choice.
� Type 2: Decision making under risk.�The decision makerdoes know the
probabilities of the various outcomes.
� Decision making under uncertainty.�The decision makerdoes not know the
probabilities of the various outcomes.
3
3-9
Decision MakingDecision Makingunder Uncertaintyunder Uncertainty
� Maximax
� Maximin
� Equally likely (Laplace)
� Criterion of realism
� Minimax
3-10
Decision Table for Decision Table for Thompson LumberThompson Lumber
0
-20,000
-180,000
Unfavorable Market ($)
0
100,000
200,000
Favorable Market ($)
State of Nature
Do nothing
Construct a small plant
Construct a large plant
Alternative
� Maximax: Optimistic Approach� Find the alternative that maximizes the maximum
outcome for every alternative.
3-11
Thompson Lumber: Thompson Lumber: Maximax SolutionMaximax Solution
000Do nothing
100,000-20,000100,000Construct a small plant
-180,000
Unfavorable Market ($)
200,000
Favorable Market ($)
State of Nature
Construct a large plant
Alternative
200,000
Maximax
3-12
Decision Table for Decision Table for Thompson LumberThompson Lumber
0
-20,000
-180,000
Unfavorable Market ($)
0
100,000
200,000
Favorable Market ($)
State of Nature
Do nothing
Construct a small plant
Construct a large plant
Alternative
� Maximin: Pessimistic Approach� Choose the alternative with maximum
minimum output.
4
3-13
Thompson Lumber: Thompson Lumber: Maximin SolutionMaximin Solution
000Do nothing
-20,000-20,000100,000Construct a small plant
-180,000
Unfavorable Market ($)
200,000
Favorable Market ($)
State of Nature
Construct a large plant
Alternative
-180,000
Maximin
3-14
Thompson Lumber: Thompson Lumber: HurwiczHurwicz
� Criterion of Realism (Hurwicz)� Decision maker uses a weighted average based
on optimism of the future.
0
-20,000
-180,000
Unfavorable Market ($)
0
100,000
200,000
Favorable Market ($)
State of Nature
Do nothing
Construct a small plant
Construct a large plant
Alternative
3-15
Thompson Lumber: Thompson Lumber: Hurwicz SolutionHurwicz Solution
CR = α*(row max)+(1- α)*(row min)
000Do nothing
76,000-20,000100,000Construct a small plant
-180,000
Unfavorable Market ($)
200,000
Favorable Market ($)
State of Nature
Construct a large plant
Alternative
124,000
Criterion of Realism
or Weighted
Average (α= 0.8) ($)
3-16
Decision MakingDecision Makingunder Uncertaintyunder Uncertainty
� Equally likely (Laplace)�Assume all states of nature to be
equally likely, choose maximum Average.
0
-20,000
-180,000
Unfavorable Market ($)
0
100,000
200,000
Favorable Market ($)
State of Nature
Do nothing
Construct a small plant
Construct a large plant
Alternative
5
3-17
Decision MakingDecision Makingunder Uncertaintyunder Uncertainty
0
-20,000
-180,000
Unfavorable Market ($)
0
100,000
200,000
Favorable Market ($)
State of Nature
0Do nothing
40,000Construct a small plant
10,000
Avg.
Construct a large plant
Alternative
3-18
Thompson Lumber;Thompson Lumber;Minimax RegretMinimax Regret
� Minimax Regret:� Choose the alternative that minimizes the
maximum opportunity loss .
0
-20,000
-180,000
Unfavorable Market ($)
0
100,000
200,000
Favorable Market ($)
State of Nature
Do nothing
Construct a small plant
Construct a large plant
Alternative
3-19
Thompson Lumber:Thompson Lumber:Opportunity Loss Opportunity Loss
TableTable
0 – 0 = 0
0- (-20,000) = 20,000
0- (-180,000) = 180,000
Unfavorable Market ($)
200,000 – 0 = 0
200,000 -100,000 = 100,000
200,000 –200,000 = 0
Favorable Market ($)
State of Nature
Do nothing
Construct a small plant
Construct a large plant
Alternative
3-20
Thompson Lumber:Thompson Lumber:Minimax Regret Minimax Regret
SolutionSolution
0
20,000
180,000
Unfavorable Market ($)
200,000
100,000
0
Favorable Market ($)
State of Nature
200,000Do nothing
100,000Construct a small plant
180,000
Maximum Opportunity
Loss
Construct a large plant
Alternative
6
3-21
InIn --Class Example 1Class Example 1
� Let’s practice what we’ve learned. Use the decision table below to compute (1) Mazimax (2) Maximin (3) Minimax regret
0
35,000
25,000
Average
Market
($)
0
-60,000
-40,000
Poor
Market
($)
0
100,000
75,000
Good
Market
($)
State of Nature
Do nothing
Construct a small plant
Construct a large plant
Alternative
3-22
InIn --Class Example 1:Class Example 1:MaximaxMaximax
0
35,000
25,000
Average
Market
($)
0
-60,000
-40,000
Poor
Market
($)
0
100,000
75,000
Good
Market
($)
State of Nature
0Do nothing
100,000Construct a small plant
75,000
Maximax
Construct a large plant
Alternative
3-23
InIn --Class Example 1:Class Example 1:MaximinMaximin
0
35,000
25,000
Average
Market
($)
0
-60,000
-40,000
Poor
Market
($)
0
100,000
75,000
Good
Market
($)
State of Nature
0Do nothing
-60,000Construct a small plant
-40,000
Maximin
Construct a large plant
Alternative
3-24
InIn --Class Example 1:Class Example 1:Minimax Regret Minimax Regret
Opportunity Loss TableOpportunity Loss Table
35,000
0
75,000
Average
Market
($)
0
60,000
40,000
Poor
Market
($)
100,000
0
25,000
Good
Market
($)
State of Nature
100,000Do nothing
60,000Construct a small plant
40,000
Maximum Opp. Loss
Construct a large plant
Alternative
7
3-25
Decision Making under Decision Making under RiskRisk
Expected Monetary Value:
In other words:EMV Alternative n = Payoff 1 * PAlt. 1 + Payoff 2
* PAlt. 2 + … + Payoff n * PAlt. n
nature. of stagesof numbern where
SP SPayoffative)EMV(Altern j
n
jj
=
=∑=
)(*1
3-26
Thompson Lumber:Thompson Lumber:EMVEMV
0.50
0
-20,000
-180,000
Unfavorable Market ($)
0.50
0
100,000
200,000
Favorable Market ($)
State of Nature
Probabilities
0*0.5 + 0*0.5 = 0Do nothing
100,000*0.5 +
(-20,000)*0.5 = 40,000
Construct a small plant
200,000*0.5 +
(-180,000)*0.5 = 10,000
EMV
Construct a large plant
Alternative
3-27
Thompson Lumber:Thompson Lumber:EV|PI and EMV EV|PI and EMV
SolutionSolution
0*0.5 = 0
0
-20,000
-180,000
Unfavorable Market
($)
200,000*0.5 =
100,000
0
100,000
200,000
Favorable Market
($)
State of Nature
EV�PI
0Do nothing
40,000Construct a small plant
10,000
EMV
Construct a large plant
Alternative
3-28
Expected Value of Expected Value of Perfect Information Perfect Information
((EVPIEVPI))� EVPI places an upper bound on what
one would pay for additional information.
� EVPI is the expected value with perfect information minus the maximum EMV.
8
3-29
Expected Value with Expected Value with Perfect Information Perfect Information
((EV|PIEV|PI))
In other words
EV �PI = Best Outcome of Alt 1 * PAlt. 1 + Best Outcome of Alt 2 * PAlt. 2 +… + Best Outcome of Alt n * PAlt. n
nature. of states ofnumber n
)P(S*nature) of statefor outcome(Best PI|EVn
1jj
=
=∑=
3-30
Expected Value of Expected Value of Perfect InformationPerfect Information
EVPI = EV|PI - maximum EMV
Expected value with perfect information
Expected value with no additional
information
3-31
Thompson Lumber:Thompson Lumber:EVPI SolutionEVPI Solution
EVPIEVPI = expected value with perfect
information - max(EMVEMV)
= $200,000*0.50 + 0*0.50 - $40,000
= $60,000 From previous slide
3-32
InIn --Class Example 2Class Example 2
Let’s practice what we’ve learned. Using the table below compute EMV, EV �PI, and EVPI.
0
35,000
25,000
Average
Market
($)
0
-60,000
-40,000
Poor
Market
($)
0
100,000
75,000
Good
Market
($)
State of Nature
Do nothing
Construct a small plant
Construct a large plant
Alternative
9
3-33
InIn --Class Example 2:Class Example 2:EMV and EVEMV and EV ��PIPI
SolutionSolution
0
35,000
25,000
Average
Market
($)
0
-60,000
-40,000
Poor
Market
($)
0
100,000
75,000
Good
Market
($)
State of Nature
0Do nothing
27,500Construct a small plant
21,250
EMV
Construct a large plant
Alternative
3-34
InIn --Class Example 2:Class Example 2:EVPI SolutionEVPI Solution
EVPIEVPI = expected value with perfect
information - max(EMVEMV)
= $100,000*0.25 + 35,000*0.50 +0*0.25
= $ 42,500 - 27,500
= $ 15,000
3-35
Expected Opportunity Expected Opportunity LossLoss
� EOL is the cost of not picking the best solution.EOL = Expected Regret
3-36
Thompson Lumber: EOLThompson Lumber: EOLThe Opportunity Loss TableThe Opportunity Loss Table
0.50
0-0
0 – (-20,000)
0- (-180,000)
Unfavorable Market ($)
0.50
200,000 - 0
200,000 -100,000
200,000 –200,000
Favorable Market ($)
State of Nature
Probabilities
Do nothing
Construct a small plant
Construct a large plant
Alternative
10
3-37
Thompson Lumber: Thompson Lumber: EOL TableEOL Table
0.50
0
-20,000
-180,000
Unfavorable Market ($)
0.50
0
100,000
200,000
Favorable Market ($)
State of Nature
Probabilities
Do nothing
Construct a small plant
Construct a large plant
Alternative
3-38
Thompson Lumber: Thompson Lumber: EOL SolutionEOL Solution
Alternative EOL
Large Plant (0.50)*$0 + (0.50)*($180,000)
$90,000
Small Plant (0.50)*($100,000)+ (0.50)(*$20,000)
$60,000
Do Nothing (0.50)*($200,000) + (0.50)*($0)
$100,000
3-39
Thompson Lumber:Thompson Lumber:Sensitivity AnalysisSensitivity Analysis
EMV(Large Plant):
= $200,000P - (1-P)$180,000
EMV(Small Plant):
= $100,000P - $20,000(1-P)
EMV(Do Nothing):
= $0P + 0(1-P)
3-40
Thompson Lumber:Thompson Lumber:Sensitivity AnalysisSensitivity Analysis(continued)(continued)
-200000
-150000
-100000
-50000
0
50000
100000
150000200000250000
0 0.2 0.4 0.6 0.8 1
Values of P
EM
V V
alue
s
Point 1 Point 2Small Plant
Large Plant EMV
11
3-41
Marginal AnalysisMarginal Analysis
� P = probability that demand >a given supply.
� 1-P = probability that demand < supply.
� MP = marginal profit.
� ML = marginal loss.
� Optimal decision rule is:
� P*MP ≥ (1-P)*ML
or
MLMPML
P++++
≥≥≥≥
3-42
Marginal Analysis Marginal Analysis --Discrete DistributionsDiscrete Distributions
Steps using Discrete Distributions:
� Determine the value forP.P.
� Construct a probability table and add a cumulative probability column.
� Keep ordering inventory as long as the probability of selling at least oneadditional unit is greater thanP.P.
3-43
Café du Donut:Café du Donut:Marginal AnalysisMarginal Analysis
Daily Sales
(Cartons)
Probability of Sales
at this Level
Probability that Sales Will
Be at this Level or Greater
4 0.05 1.00
5 0.15 0.95
6 0.15 0. 80
7 0.20 0.65
8 0.25 0.45
9 0.10 0.20
10 0.10 0.10
1.00
Café du Donut sells a dozen donuts for $6. It costs $4 to make each dozen. The following table shows the discrete distribution for Café du Donut sales.
3-44
Café du Donut: Café du Donut: Marginal Analysis SolutionMarginal Analysis Solution
Marginal profit = selling price - cost
= $6 - $4 = $2Marginal loss = cost
Therefore:
667.06
4
24
4 ==+
=
+≥
MPML
MLP
12
3-45
Café Café dudu Donut: Donut: Marginal Analysis SolutionMarginal Analysis Solution
Daily Sales
(Cartons)
Probability of Sales
at this Level
Probability that Sales Will
Be at this Level or Greater
4 0.05 1.00 ≥ 0.66
5 0.15 0.95 ≥ 0.66
6 0.15 0. 80 ≥ 0.66
7 0.20 0.65
8 0.25 0.45
9 0.10 0.20
10 0.10 0.10
1.00
3-46
Daily Sales Cases
Probability of Sales at this Level
Probability that Sales Will Be at this
Level or Greater 4 0.1
5 0.1 6 0.4
7 0.3 8 0.1
1.00
InIn --Class Example 3Class Example 3
Let’s practice what we’ve learned. You sell cases of goods for $15/case, the raw materials cost you $4/case, and you pay $1/case commission.
3-47
InIn --Class Example 3:Class Example 3:SolutionSolution
Daily Sales Cases
Probability of Sales at this Level
Probability that Sales Will Be at this
Level or Greater 4 0.1 1.0 > .286 5 0.1 .9 > .286 6 0.4 .8 > .286 7 0.3 .4 > .286 8 0.1 .1 1.00
MP = $15-$4-$1 = $10 per case ML = $4P>= $4 / $10+$4 = .286
3-48
Marginal AnalysisMarginal AnalysisNormal DistributionNormal Distribution
�� µµ = average or mean sales
�� σσ = standard deviation of sales
�� MPMP = marginal profit
�� MLML = Marginal loss
13
3-49
Marginal Analysis Marginal Analysis --Discrete DistributionsDiscrete Distributions
• Steps using Normal Distributions:� Determine the value forP.
� LocateP on the normal distribution. For a given area under the curve, we find Z from the standard Normal table.
� Using we can now solve for:
σσσσ
µµµµ−−−−====
*XZ
MPML
MLP
++++====
X*
3-50
Marginal Analysis:Marginal Analysis:Normal Curve ReviewNormal Curve Review
σµ−=
=←
*
00.1
xZ
Pcumulative
µ Zo+Zo−
3-51
Marginal Analysis Marginal Analysis --Normal Curve ReviewNormal Curve Review
*Xµ
area = .30
Use table to find Z
area = .70
MPML
ML.3
++++====
3-52
JoeJoe’’ s Newsstand s Newsstand ExampleExample
Joe sells newspapers for $1.00 each.
Papers cost him $.40 each. His average
daily demand is 50 papers with a standard
deviation of 10 papers. Assuming sales
follow a normal distribution, how many
papers should Joe stock?
�� MLML = $0.40
�� MPMP = $0.60
�� µµ = Average demand = 50 papers per day
�� σσ = Standard deviation of demand = 10
14
3-53
JoeJoe’’ s Newsstand Examples Newsstand Example(continued)(continued)
Step 1: 404040404040404000000000.60.60.60.60.60.60.60.60.40.40.40.40.40.40.40.40
.40.40.40.40.40.40.40.40..
MPMPMLML
MLMLPP ========
++++++++========
++++++++========
.
3-54
JoeJoe’’ s Newsstand Examples Newsstand Example(continued)(continued)
Step 2: Look on the Normal table for
PP = 0.6 (i.e., 1 - .4) ∴∴∴∴ ZZ = 0.25,
and
or: 1010101010101010
5050505050505050252525252525252500000000
−−−−−−−−========
**XX
XX ** = 10 * 0.25 + 50 = 52.5 or 53 newspapers= 10 * 0.25 + 50 = 52.5 or 53 newspapers
3-55
JoeJoe’’ s Newsstand s Newsstand Example BExample B
� Joe also offers his clients the “Times” for $1.00. This paper is flown in from out of state, which greatly increases its costs. Joe pays $.80 for the “Times.”The “Times” has average daily sales of 100 papers with a standard deviation of 10. Assuming sales follow a normal distribution, how many “Times”papers should Joe stock?
�� MLML = $0.80
�� MPMP = $0.20
�� µµ = Average demand = 100 papers per day
�� σσ = Standard deviation of demand = 10
3-56
JoeJoe’’ s Newsstand s Newsstand Example BExample B (continued)(continued)
Step 1: 808080808080808000000000.8.8.8.8.8.8.8.8 .2.2.2.2.2.2.2.2
.8.8.8.8.8.8.8.8..
MPMPMLML
MLMLPP ========
++++++++========
++++++++========
.
15
3-57
Step 2:
Z = 0.80
= -0.84 for an area of 0.80
And
or: X=-8.4+100 or 92 newspapers1010101010101010
100100100100100100100100.84.84.84.84.84.84.84.8400000000
−−−−−−−−========−−−−−−−−
**XX
JoeJoe’’ s Newsstand s Newsstand Example BExample B (continued)(continued)
3-58
Decision Making with Decision Making with Uncertainty: Using the Uncertainty: Using the
Decision TreesDecision TreesDecision treesDecision trees enable one to look at
decisions:
� With many alternativesalternatives and states states
of nature,of nature,
� which must be made in sequence.
3-59
Five Steps toFive Steps toDecision Tree AnalysisDecision Tree Analysis
1. Define the problem.
2. Structure or draw the decision tree.
3. Assign probabilities to the states of nature.
4. Estimate payoffs for each possible combination of alternatives and states of nature.
5. Solve the problem by computing expected monetary values (EMVs) for each state of nature node.
3-60
Structure of Decision Structure of Decision TreesTrees
A graphical representation where:
� A decision node from which one of several alternatives may be chosen.
� A state-of-nature node out of which one state of nature will occur.
16
3-61
ThompsonThompson’’ s Decision s Decision TreeTree
1
2
A A Decision Decision
NodeNode
A State of A State of Nature Nature NodeNode
Favorable Market
Unfavorable Market
Favorable Market
Unfavorable Market
Construct
Large Plant
Construct Small Plant
Do Nothing
Step 1: Define the problem
Lets re-look at John Thompson’s decision regarding storage sheds. This simple problem can be depicted using a decision tree.
Step 2: Draw the tree
3-62
ThompsonThompson’’ s Decision s Decision TreeTree
1
2
A A Decision Decision
NodeNode
A State of A State of Nature NodeNature Node Favorable (0.5)
Market
Unfavorable (0.5)Market
Favorable (0.5)Market
Unfavorable (0.5)Market
Construct
Large Plant
Construct Small Plant
Do Nothing
$200,000$200,000
--$180,000$180,000
$100,000$100,000
--$20,000$20,000
00
Step 3: Assign probabilities to the states of nature.
Step 4: Estimate payoffs.
3-63
ThompsonThompson’’ s Decision s Decision TreeTree
1
2
A Decision A Decision NodeNode
A State A State of Nature of Nature NodeNode Favorable (0.5)
Market
Unfavorable (0.5)Market
Favorable (0.5)Market
Unfavorable (0.5)Market
Construct
Large Plant
Construct Small Plant
Do Nothing
$200,000$200,000
--$180,000$180,000
$100,000$100,000
--$20,000$20,000
00
EMV EMV =$40,000=$40,000
EMVEMV=$10,000=$10,000
Step 5: Compute EMVs and make decision.
3-64
Thompson’s Decision:Thompson’s Decision:A More Complex A More Complex
ProblemProblem� John Thompson has the opportunity of
obtaining a market survey that will give additional information on the probable state of nature. Results of the market survey will likely indicate there is a percent change of a favorable market. Historical data show market surveys accurately predict favorable markets 78 % of the time. Thus P(Fav. Mkt / Fav. Survey Results) = .78
� Likewise, if the market survey predicts an unfavorable market, there is a 13 % chance of its occurring. P(Unfav. Mkt / Unfav. Survey Results) = .13
� Now that we have redefined the problem (Step 1), let’s use this additional data and redraw Thompson’s decision tree (Step 2).
17
3-65
ThompsonThompson’’ s Decision s Decision TreeTree
3-66
ThompsonThompson’’ s Decision s Decision TreeTree
Step 3: Assign the new probabilities to the states of nature.
Step 4: Estimate the payoffs.
3-67
Thompson’s Decision Thompson’s Decision TreeTree
Step 5: Compute the EMVs and make decision.
3-68
John Thompson DilemmaJohn Thompson Dilemma
John Thompson is not sure how much value to place on market survey. He wants to determine the monetary worth of the survey. John Thompson is also interested in how sensitive his decision is to changes in the market survey results. What should he do?
�Expected Value of Sample Information
�Sensitivity Analysis
18
3-69
Expected Value of Expected Value of Sample InformationSample Information
Expected value of best decision withwith sample information, assuming no cost to gather it
Expected value of best decision withoutwithout sample information
EVSIEVSI =
EVSI for Thompson Lumber = $59,200 - $40,000
= $19,200Thompson could pay up to $19,200 and come out ahead.
3-70
Calculations for Thompson Calculations for Thompson Lumber Sensitivity Lumber Sensitivity
AnalysisAnalysis
2,400$104,000
($2,400)($106,400)1)EMV(node
++++====
−−−−++++====
p
)p(p 1111
Equating the EMVEMV(node 1) to the EMV of not conducting the survey, we have
0.36$104,000
$37,600
or
$37,600$104,000
$40,000$2,400$104,000
========
====
====++++
p
p
p
3-71
InIn --Class Problem 3Class Problem 3
Let’s practice what we’ve learned
Leo can purchase a historic home for $200,000 or land in a growing area for $50,000. There is a 60% chance the economy will grow and a 40% change it will not. If it grows, the historic home will appreciate in value by 15% yielding a $30,00 profit. If it does not grow, the profit is only $10,000. If Leo purchases the land he will hold it for 1 year to assess the economic growth. If the economy grew during the first year, there is an 80% chance it will continue to grow. If it didnot grow during the first year, there is a 30% chance it will grow in the next 4 years. After a year, if the economy grew, Leo will decide either to build and sell a house or simply sell the land. It will cost Leo $75,000 to build a house that will sell for a profit of $55,000 if the economy grows, or $15,000 if it does not grow. Leo can sell the land for a profit of $15,000. If, after a year, the economy does not grow, Leo will either develop the land, which will cost $75,000, or sell the land for a profit of $5,000. If he develops the land and theeconomy begins to grow, he will make $45,000. If he develops the land and the economy does not grow, he will make $5,000.
3-72
InIn --Class Problem 3: Class Problem 3: SolutionSolution
1
2
3
4
5
6
7
Purchase historic home
Purchase land
Economy grows (.6)
No growth (.4)
Economy grows (.6)
No growth (.4)
Build house
Economy grows (.8)
No growth (.2)
Sell land
Develop land
Sell land
Economy grows (.3)
No growth (.7)
19
3-73
InIn --Class Problem 3: Class Problem 3: SolutionSolution
1
2
3
4
5
6
7
Purchase historic home
Purchase land
$35,000
$22,000 Economy grows (.6) $30,000
No growth (.4)
$10,000
Economy grows (.6)
No growth (.4)
$35,000
$47,000
Build house
$47,000
Economy grows (.8) $55,000
$15,000No growth (.2)
Sell land
$15,000
$17,000
Develop land
Sell land
$5,000
Economy grows (.3)
No growth (.7)
$45,000
$5,000
$17,000
3-74
Estimating Probability Estimating Probability Values with BayesianValues with Bayesian
� Management experience or intuition
� History
� Existing data
� Need to be able to reviseprobabilities based upon new data
Posteriorprobabilities
Priorprobabilities New data
Baye’s Theorem
3-75
Bayesian AnalysisBayesian Analysis
Market Survey Reliability in Predicting Actual States of Nature
Actual States of Nature
Result of Survey Favorable
Market (FM)
Unfavorable
Market (UM)
Positive (predicts
favorable market
for product)
P(survey positive|FM)
= 0.70
P(survey positive|UM)
= 0.20
Negative (predicts
unfavorable
market for
product)
P(survey
negative|FM) = 0.30
P(survey negative|UM)
= 0.80
The probabilities of a favorable / unfavorable state of nature can be obtained by analyzing the Market Survey Reliability in Predicting Actual States of Nature.
3-76
Bayesian Analysis Bayesian Analysis (continued):(continued):Favorable SurveyFavorable Survey
Probability Revisions Given a Favorable Survey
Conditional
Probability
Posterior
Probability
State
of
Nature
P(Survey positive|State of Nature
Prior ProbabilityJoint Probability
FM 0.70 * 0.50 0.350.45
0.35= 0.78
UM 0.20 * 0.500.45
0.100.10 = 0.22
0.45 1.00
20
3-77
Bayesian Analysis Bayesian Analysis (continued):(continued):Unfavorable SurveyUnfavorable Survey
Probability Revisions Given an Unfavorable Survey
Conditional
Probability
Posterior
Probability
State
of
Nature
P(Survey
negative|State
of Nature)
Prior Probability
Joint Probability
FM 0.30 * 0.50 0.150.55
0.15= 0.27
UM 0.80 * 0.50 0.400.55
0.40= 0.73
0.551.00
3-78
Decision Making Using Decision Making Using Utility TheoryUtility Theory
� Utility assessment assigns the worst outcome a utility of 0, and the bestoutcome, a utility of 1.
� A standard gamble is used to determine utility values.
� When you are indifferent, the utility values are equal.
3-79
Standard Gamble for Standard Gamble for Utility AssessmentUtility Assessment
Best outcomeUtility = 1
Worst outcomeUtility = 0
Other outcomeUtility = ??
(p)
(1-p)Alternativ
e 1
Alternative 2
3-80
Simple Example: Utility Simple Example: Utility TheoryTheory
$5,000,000
$0
$2,000,000
Accept Offer
Reject Offer
Heads(0.5)
Tails(0.5)
Let’s say you were offered $2,000,000 right now on a chance to win $5,000,000. The $5,000,000 is won only if you flip a coin and get tails. If you get heads you lose and get $0. What should you do?
21
3-81
Real Estate Example: Real Estate Example: Utility TheoryUtility Theory
Jane Dickson is considering a 3-year real estate investment. There is an 80 % chance the real estate market will soar and a 20 % chance it will bust. In a good market the real estate investment will pay $10,000, in an unfavorable market it is $0. Of course, she could leave her money in the bank and earn a $5,000 return. What should she do?
3-82
Real Estate Example: Real Estate Example: SolutionSolution
$10,000U($10,000) = 1.0
0U(0)=0
$5,000U($5,000)=p=0.80
p= 0.80
(1-p)= 0.20
Invest in
Real Estat
e
Invest in Bank
3-83
Utility Curve for Jane Utility Curve for Jane DicksonDickson
00.10.20.30.40.50.60.70.80.9
1
$- $2,000 $4,000 $6,000 $8,000 $10,000
Monetary Value
Uti
lity
3-84
Preferences for RiskPreferences for Risk
Monetary Outcome
Risk
Avoider
Risk
Seeke
r
Risk In
differ
ence
Util
ity
22
3-85
Decision Facing Mark Decision Facing Mark SimkinSimkin
Tack landspoint up(0.45)
Tack landspoint down (0.55)
$10,000
-$10,000
0
Alternativ
e 1
Mark play
s
the gam
e
Alternative 2
Mark does not play the game
3-86
Utility Curve for Mark Utility Curve for Mark SimkinSimkin
0
0.1
0.20.3
0.4
0.5
0.6
0.70.8
0.9
1
-$20,000 -$10,000 $0 $10,000 $20,000 $30,000
