Decision Analysis
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Transcript of Decision Analysis
DECISION ANALYSIS
Decision analysis addresses the issue of making the right decision in the face of great uncertainty.It provides a framework and methodology for rational decision making when the outcomes are uncertain.
CASES
1. A manufacturer introducing a new product into the market place. Questions are:• How much should be
produced?• Should the product be test
marketed?• How much advertising?
• What are the market sectors and individual securities with the best prospects?
• Where is the economy heading?
• How will interest rates behave?• How should these factors affect
the investment decisions?
2. A financial firm investing in securities:
3. A government contractor bidding on a new contract:
• What other companies are bidding?
• What are there likely bids?• What will be cost of the
project?
4. An agricultural company selecting the mix of crops:• What will be the weather
conditions?
• Where are prices headed?
• What will costs be?5. An oil company deciding to drill for
oil at a particular location: • How likely is oil there?
• How much?
• How deep to drill?
• Should Geologists do further investigation before drilling?
In all these cases the decision maker has to answer his questions and arrive at the right decision when the environment has uncertainty. Decision analysis is used to answer questions in these types of scenarios.
Decision analysis is divided into
two types:
1. Decision making without
experimentation
2. Decision making with
experimentation
In the first case decision-
making is done immediately.
In the second case decision-
making is done after some
testing is done to reduce the
level of uncertainty.
Framework for decision analysis
1. The decision maker must choose an action from a set of possible actions.
2. Different situations would be there when the action is undertaken. Each of these situations is called a state of nature.
3. Each combination of action and state of nature would give rise to some monetary gain. These are known as payoffs.
4. A payoff table is used to provide the payoff for each combination.
5. The decision maker will also have some information about the likelihood of a state of nature occurring. These are known as prior probabilities.
Example
An oil company ‘A’ owns a tract of land that may contain oil. A Geologist has told management that there is 1 chance in 4 that it contain oil. Another oil company has offered to purchase the land for $90,000. Company ‘A’ is considering drilling for oil itself. The cost of drilling is $100,000. Expected revenue is $800,000, Expected profit is $700,000. A loss of $100,000 will be incurred if the land contains no oil.
Profit Table
PAYOFF
ALTERNATIVE OIL DRY
Drill for Oil $700,000 -$100,000
Sell the Land $90,000 $90,000
Chance 25% 75%
Payoff Table
STATE OF NATURE
ALTERNATIVE OIL DRY
Drill for Oil 700 -100
Sell the Land 90 90
Prior Probability 0.25 0.75
Decision Analysis without experimentation has three criterions by which to select the action to be undertaken:
1. The maxmin payoff criterion
2. The maximum likelihood criterion
3. Bays decision rule
1. The Maximin Pay Off Criterion
For each possible action, find the minimum payoff over all possible states of nature. Find the maximum of these minimums. Choose the action whose minimum payoff gives this maximum.
State of Nature
ALTERNATIVE OIL DRY MINIMUM
Drill 700 -100 -100
Sell 90 90 90 maximum
Under this criterion for the example the action to be taken is to sell the land.
2. Maximum Likelihood Criterion
Identify the most likely state
of nature (the one with the
largest prior probability). For
this state of nature, find the
action with the maximum
payoff. Choose this action: State of Nature
ALTERNATIVE OIL DRY MAXIMUM
Drill 700 -100
Sell 90 90 Maximum
0.25 0.75
3. Bayes Decision Rule
Using the prior probability calculate the expected value of the payoff for each of the possible actions. Choose the action with the maximum expected payoff. E [Payoff (Drill)]
= 0.25 (700) + 0.75 (-100) = 100
E [Payoff (Sell)]
= 0.25 (90) + 0.75 (90) = 90
State Of Nature
ALTERNATIVE OIL DRY EXPECTEDPAYOFF
Drill 700 -100 100 Maximum
Sell 90 90 90
Prior Probability 0.25 0.75
CLASS WORK
There are 3 courses of action with 3 state of nature, as follows: State Of Nature
Alternative Improving Economy
StableEconomy
Worsening Economy
Conservative Investment
$30M $5M &10M
Speculative Investment
&40M $10M -$30M
CounterCyclical
-$10M10.1
00.5
$15M0.4
Which action should betaken under:1. Maximum Payoff Criteria2. Maximum Likelihood3. Bayes Decision Rule
CLASS WORK 1. A manager of a Grocery store
needs to replenish her supply of apples. Her regular supplier can provide as many cases as she wants. However, because these apples already are very ripe, she will need to sell them tomorrow and discard any that remain unsold. The manager estimates that she will be able to sell 10, 11, 12 or 13 cases tomorrow. She can purchase the apples for $3 per cases and sell them for $8 per case. The manager now needs to decide how many cases to purchase. The prior probabilities are 0.2,0.4, 0.3 and 0.1 of being able to sell 10, 11, 12 and 13 cases.
a. Develop a decision analysis formulation of this problem
b. How many cases of apples should be purchased under the maximin payoff criterion?
c. How many cases should be purchased according to the maximum likelihood criterion?
d. How many cases should be purchased according to bayes decision rule?
2. Consider a decision analysis problem whose payoffs are:
ALTERNATIVE STATE OF NATURE
S1 S2
A1 80 25
A2 30 50
A3 60 40
Prior Probabilities 0.4 0.6
a. Which alternative should be chosen under the maximum payoff criterion
b. Which alternative should be chosen under maximum likelihood criterion
c. Which alternative should be chosen under bayes decision rule
3. The following is a payoff table for a decision analysis problem:
ALTERNATIVE STATE OF NATURE
S1 S2 S3
A1 220 170 110
A2 200 180 150
Prior Probabilities
0.6 0.3 0.1
a. Which alternative should be chosen under the maxmin payoff criterion?
b. Which alternative should be chosen under the maximum likelihood criterion?
c. Which alternative should be chosen under bayes decision rule?
4. There is a manager of a large farm with 1000 acres of arable land. For greater efficiency, the manager always devotes the farm to growing one crop at a time. He now needs to make a decision on which one of four crops to grow. For each of these crops, the manager has obtained the following estimates of crop yields and net incomes per bushel under various weather conditions.
Weather Crop 1 Crop 2 Crop 3 Crop 4
Dry 20 15 30 40
Moderate 35 20 25 40
Damp 40 30 25 40
Net income
per bushel
$1.00 $1.50 $1.00 $0.50
The prior probabilities are 0.3, 0.5, 0.2 for dry, moderate, damp weathera. Develop a decision analysis
formulation of this problem.
b. Determine which crop to grow using Bayes Decision Rule
Decision analysis with experimentation
Many times additional testing can be done to improve the prior probabilities of the states of nature. These improved estimates are called posterior probabilities.Notations
n = Number of possible states of nature
P (State = State i) = Prior probability for state of nature i i = 1, 2,3 …. n.
Finding = Finding from experimentation or testing
Finding j = One value of finding
P [State = State i | Finding = Finding j] = posterior probability that, state of nature is i., given that finding = finding j.
To obtain posterior probabilities we are given;
P (State = State i) and P (Finding = Finding j| State = State i) .
For each I = 1, 2, 3, …………n the formula for the posterior probability is
P(STATE = STATE i FINDING = FINDING j)
n
k
STATEkSTATEPSTATEkSTATEFINDINGjFINDINGP
STATEiSTATEPSTATEiSTATEFINDINGjFINDINGP
1
)()(
)()(
Example:
For Oil Company ‘A’ there is an option of carrying out extra testing or a detailed seismic survey of the land. This will enable the company to obtain an improved estimate of oil. The cost of doing the survey is $30, 000.The findings from the survey can be divided into two categories:
USS: Unfavorable Seismic Soundings, oil is unlikely
FSS: Favorable Seismic Sounding, oil is likely.
Based on past experience
P (USSSTATE = OIL) = 0.4
P (FSS STATE = OIL) = 0.6
P (USS STATE = DRY) = 0.8
P (FSS I STATE = DRY = 0.2
PRIOR PROBABILITIES
P (STATE=OIL) = 0.25, P (STATE =DRY) = 0.75
P (STATE = OIL FINDING = USS)
14.0)75.0(8.0)25.0(4.0
)25.0(4.0
P (STATE = DRY FINDING = USS) =
86.0)75.0(8.0)25.0(4.0
)25.0(8.0
P (STATE = OIL FINDING = FSS) =
P (STATE = DRY FINDING = FSS) =
5.0)75.0(2.0)25.0(6.0
)25.0(6.0
5.0)75.0(2.0)25.0(6.0
)75.0(2.0
We next apply Bayes Decision Rule to obtain the expected payoffs
PROBABILITIES STATEFINDING P (FINDING) OIL DRY
FSS 0.3 0.5 0.5
USS 0.7 0.14 0.86
All these calculations can be organized in a Probability Tree Diagram
DRY & USS
0.25(0.6) = 0.15
0.75(0.8) = 0.6
0.75(0.2) = 0.18
0.25(0.4) = 0.1
OIL & FSS
OIL & USS
DRY & FSS
DRY, GIVEN USS
DRY, GIVEN FSS
OIL, GIVEN USS
OIL, GIVEN FSS
0.15/0.3 = 0.5
0.1/0.7 = 0.14
0.15/0.3 = 0.5
0.6/0.7 = 0.86
Prior Probabilities
Conditional Probabilities Joint Probabilities Posterior Probabilities
P (FINDINGSTATE) P (STATE AND FINDING) P(STATEFINDING)
After the calculations are done, Bayes Decision Rule can be applied with the posterior probabilities replacing the prior probabilities
Payoff Table: Alternative Oil Dry
1. Drill 700 - 100
2. SellUSSFSS
900.140.5
900.860.5
If finding is unfavourable:
E [Payoff (Drill)Finding = USS]
= 0.14(700) + 0.86(- 100) – 30 = - 18
E [Payoff (Sell)Finding = USS]
= 0.14(90) + 0.86(90) – 30 = 60If finding is favourable:
E [Payoff (Drill)Finding = FSS]
= 0.5(700) + 0.5(-100) – 30 = 270
E [Payoff (Sell)Finding = FSS]
= 0.5(90) + 0.5(90) – 30 = 60
Finding Action Expected Payoff
USS Sell 60
FSS Drill 270
Class Work
For the oil company example a consulting geologist has given more precise estimates on the likelihood of obtaining favorable seismic soundings. Specifically when the land contains oil, favorable seismic soundings are obtained 80% of the time. When the land is dry, favourable seismic soundings are obtained 40% of the time.
1. What are the posterior probabilities, show in a probability tree diagram?
Solution
1.
DRY & USS
0.2
0.45
0.3
0.05
OIL & FSS
OIL & USS
DRY & FSS DRY, GIVEN FSS
OIL, GIVEN USS
OIL, GIVEN FSS
0.4
0.1
0.6
0.9DRY, GIVEN USS
= (0.25)(0.8)
0.45/0.5
0.05/0.5
02/05
0.3/0.5
2. The optimal policy is to do a seismic survey and sell if it is unfavourable and drill if it is favourable.
Class Work – 2
The following Payoff Table is givenAlternative State of nature
S1 S2
A1 400 -100
A2 0 100
Price Prob. 0.4 0.6
Can pay $100 to have research done to better predict which state of nature will occur. When the true state of nature is S1,
the research will accurately predict S1,
60% of the time, (but will inaccurately predict S2, 40% of the time). When the
true state of nature is S2, the research will
accurately predict S2, 80% of the time (but
will inaccurately predict S1 20% of the
time
Given that research is done, determine the posterior probabilities of the states of nature for each of the two cases.
1. P(S1 and S1) = 0.4 x 0.6 = 0.24P(S1 and S2) = 0.4 x 0.4 = 0.16P(S2 and S1) = 0.5 x 0.2 = 0.12P(S2 and S2) = 0.6 x 0.8 = 0.48P(S1) = 0.24 + 0.12 = 0.36P(S2) = 0.16 + 0.48 = 0.64
S2 and S2
0.24 = 0.677
0.25
0.75
S1S10.6
S1 and S2
0.16/0.64
S1 & S1
0.24/0.26 P(S1 S1)
0.4
0.333
S2, S2 0.8
0.16P(S1 S2)
P(S2 S1)
P(S2 S2)
S1
0.2S2
S2,S1 0.12
S2 & S1
0.12/0.36
0.48/0.640.48
Example
There is a manager of a fabric mill. He is currently faced with the question of whether to extend $100,000 credit to a new customer. He has three categories for the credit-worthiness of a company, poor risk, average risk, good risk. He does not know which category this new customer fits. Experience indicates that 20% of companies similar to this dress manufacturer are poor risks, 50% are average risks and 30% are good risks. If credit is extended the expected profit for poor risks is - $15,000 for average risk $10,000 and $20,000 for good risks.
The manager is able to consult a consult a credit rating organization for a fee of $5000 per company evaluated. For companies whose actual credit record with the mill turns out to fall into each of the three categories, the following table shows the percentages that were given, each of the three possible credit evaluations by the credit rating organization.
Actual Credit Record
State
Credit Evaluation
Poor Average Good
Poor Finding 50% 40% 20%
Average 40% 50% 40%
Good 10% 10% 40%
b. Assume the credit rating organization is not used. Use Bayes Decision Rule to determine which decision alternative should be chosen.
c. Assume now that the credit rating organization is used. Develop a probability Tree Diagram to find the Posterior Probabilities of the respective states of nature. For each of the three possible credit evaluations of this potential customer.
a. Develop a decision analysis formation of this programme
Solution
State
Alternative Poor Average Good
Extend credit -15000 10000 20000
Do not extend credit
0 0 0
Prior probabilities 0.2 0.5 0.3
a.
b.Alternative Poor Average Good
Extend credit -15000 10000 20000 8000
Maximum
0
Do not extend credit
0 0 0
Prior probabilities
0.2 0.5 0.3
Extend credit (Expected payoff is $8000
3.
0.6316
0.1778
0.1053
0.5556
0.5556
0.2632
0.1657
0.2667
PS GF
PS PF
PS AF
AS GF
AS AF
AS PF
GS PF
GS AF
GS GFGS, GF
GS & AF
GS & PF
AS & GF
AS & AF
AS & PF
PS & GF
PS & AF
PS & PF0.08
0.02
0.25
0.05
0.06
0.12
0.12
0.02
PF, PS
PS
GS
GF, GS
AF, PSGF, PS
PF, AS
GF, AS
PF, GS
AF, GS0.4
0.40. 2
0.3
0.1
0.50.4
0.1
0.40.5
0.2
0.5
AS AF, AS
PF = Poor Finding PE = Poor State
AF = Average Finding AS = Average State0.1
GF = Good Finding GS = Good State
0.28
DECISION TREES
Decision trees provide away of visually displaying the decision analysis problemExample
For the oil company ‘A’ problem the decision tree is
a
c
Oil
Dryf
d
Oil
Dryg
e
Oil
Dryh
bSeismic
Unfavor
Sell
Sell
Sell
Sell
Drill
Drill
Drill
Favor
No Seismic Survey
The nodes of the Decision Tree are referred to as forks and the arcs are called branches.
Forks are of two types
1. Decision Fork – Represented by a square, indicates a decision needs to be made at that point in process
2. Chance Fork – Represented by a circle, indicates that a random event occurs at that point.
Path required to reach a terminal branch is determined both by the decision made and by random events.
a
c
670
- 130f
d
670
- 130g
e
700
- 100h
bSeismic
Unfavor (0.7)
Sell
Sell
Drill
Drill
Drill
Favor 0.3
No Seismic
90
60
0
90
- 100
- 100
- 100
- 30
Oil (0.143)
800
Dry (0.857)
90
Oil (0.5)
Oil (0.25)
Dry (0.5)
Dry (0.75)
90
60
8000
8000
Decision Tree after adding both probabilities and pay offs.
Analysis1. Start at the last column of the tree
and move left one column at a time. For each column do either step 2 or step 3 depending upon whether the Fork is a Chance Fork or a Decision Fork.
2. For each Chance Fork, calculate the expected payoff of each branch by the probability of that branch and summing these products. Record this payoff next to the relevant Fork
3. For each Decision Fork, compare the expected payoffs of its branches and choose the alternative whose branch has the largest expected payoff, cut out each rejected branch by inserting a double dash.
Example
Consider oil company example with Chance Fork f, g, h. Apply step 2.
For Fork f
EP = 0.143(670) + 0.857(-130) = -15.7
For Fork g
EP = 0.5(670) + 0.5(-130) = 270
For Fork h
EP = 0.25(700) + 0.75(-100) = 100
Moving one step to the left brings us to Decision Forks c,d and e.For Fork c
Drill Alternative has EP = - 15.7
Sell Alternative has EP = 60
60 > -15.7 so choose Sell Alternative
Fork d
Drill Alternative has EP = 270Sell Alternative has EP = 602.70 > 60, so choose Drill Alternative
Fork e
Drill Alternative has EP = 100Sell Alternative has EP = 90100 > 90, so choose Drill AlternativeWe move one more column to the left
For Fork b
EP = 0.7(60) + 0.3(270) = 123Moving one column to the left
Fork a
Do Seismic Survey has EP = 123No Seismic Survey has EP = 100123 > 100, so choose Do Seismic Survey.
Final Tree
a
c
670
- 130f
d
670
- 130g
e
700
- 100h
bSeismic
Unfavor (0.7)
Sell
270Sell
Sell
Drill
Drill
Drill
Favor 0.3
No Seismic Survey
90
60
60
90
- 100
- 100
- 100
- 30
Oil (0.143)
Dry (0.857)
90
Oil (0.5)
Oil (0.25)
Dry (0.5)
Dry (0.75)
90
800
8000
0
0
100
270
- 15.7
100
100
60
123
0
Class Work
a
c
d
b
750
2500
- 700
900
800
0.6
0.8
0.2
0.4
1. Do the analysis to obtain the final tree
Solution
- 700
a
c
b
d
820
0.2
820
750
800
900
580
2500506
900
0.6
0.4
0.8
Class Work
a
b
c
e
g
d
f
0.4
0.6
0.3
0.4
0.5
0.5
0.5
0.5
10
-10
40
-5
-10
30
10
0
1. Do the analysis
8
8
5
5
0.3
0.7
10
- 10
40
- 5
- 10
30
0
10
0.5
0.5
0.5
0.5
2.5
15
15
Solution
Class work
On Monday a certain stock closed at $10 per share, on Tuesday you expect the stock to close at $9, $10,$11 per share with respective probabilities 0.3, 0.3 and 0.4 on Wednesday, you expect the stock to close 10% lower, unchanged or 10% higher than Tuesday’s close with the following probabilities
Tuesday’s Close
10% lower
Unchanged 10% higher
$ 9$ 10$ 11
0.40.20.1
0.30.20.2
0.30.60.7
On Tuesday, you are directed to buy 100 shares of the stock before Thursday. All purchases are made at the end of the day, at the known closing price of the day, so your options are to buy at the end of Tuesday or at the end of Wednesday. You wish to determine the optimal strategy for whether to buy on Tuesday or defer the purchase until Wednesday to minimize the expected purchase price.
1. Develop and analyze a decision free for determining the optimal strategy.
Solution0.4
10% lower
0.3Unchanged
0.310% Higher
0.110% lower
0.2Unchanged
0.710% Higher
Wait
Buy
Wait
- 891
- 1168
- 891
0.210% lower
0.2Unchanged
0.610% Higher
Buy
Wait
- 1040
- 1000
- 1100
Close at $ 9
Close at $ 11
Close at $ 10
0.3
0
0.3
0.4
- 900
- 810
- 900
- 990
- 1000
- 900
- 1000
- 1100
- 1000
- 1210
- 1100
- 990
The optimal policy is to wait until Wednesday If the price is $ 9 on Tuesday. If the price is $10 or $ 11 on Tuesday than buy on Tuesday