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