1. Chapter 23
Decision Analysis

2. 23.1 Introduction
We discuss the problem of selecting an alternative from a set of possible decisions.
The decision may or may not depend on statistical evidence.
Financial considerations are explicitly included in the decision process

3. 23.2 Decision Problem
To cover several new concepts and use terminology typical to decision analysis we need to define the following:
Acts
States of nature
Payoff table
Opportunity loss
Additional definitions are mentioned later.

4. Acts Example 23.1 A man wants to invest $1 million for one year.
Three investment alternatives considered (the acts) are:
a1: guaranteed income certificate paying 10%
a2: bond with a coupon value of 8%
a3: well diversified portfolio of stocks

5. States of Nature
Return on investment in the diversified portfolio depends on the behavior of interest rate next year.
Interest rate during the year is expected to behave along one of the following three lines:
s1: Interest rate increases
s2: Interest rate stay the same
s3: Interest rater decrease.

6. Payoff Table States of Nature Acts a1 a2 a3 s1 s2 s3
100,000
-50,000
150,000 s2
80,000
90,000 s3
180,000
40,000
The entry in each cell is the payoff for a given act when a certain state of nature takes place.

7. Opportunity Loss Table
States of Nature
Acts a1 a2 a3 s1
50,000
200,000
150K-150K
150K-100K
150K-(-50K)
s , ,000
States of Nature
Acts a1 a2 a3 s1
100,000
-50,000
150,000 s2
80,000
90,000 s3
180,000
40,000

8. Opportunity Loss Table
States of Nature
Acts a1 a2 a3 s1
50,000
200,000 S2
20,000
10,000 S3
100,000
140,000

9. Decision Trees
When decisions are made sequentially, a decision tree is used to identify the best sequence.
A decision tree consists of
Branches
Nodes
Decision trees are used for the analysis of multistage decision problems

10. Decision Trees Example 23.1 – continued
Draw the decision tree that represents this decision problem.
$100,000 a1
A decision node a2
A state of nature is
about to occur a3

11. Decision Trees Example 23.1 – continued
Draw the decision tree that represents this decision problem.
$100,000 a1
A decision node
-$50,000 s1
$150,000 a2
$80,000 s2
$90,000 a3
$180,000 s3
$40,000

12. Expected Monetary Value Decision
In many decision problems we can assign probabilities to state of nature.
In theses cases decisions can be made using the expected monetary value (EMV) approach.

13. Expected Monetary Value Decision
The procedure:
For each act ai calculate the expected payoff as follows: EMV(ai) = P(s1)Payoff(ai,s1)+P(s2)Payoff(ai,s2)+… P(ai,)Payoff(ai,sk)
Choose the decision a* with the most favorable expected value.

14. Expected Monetary Value Decision; Example
Example 23.1 – continued
Use the EMV criterion to find the optimal act for the investment problem presented in this example
EMV(a1) = .2(100K) + .5(100K) + .3(100K) = 100K
EMV(a2) = .2(-50K) + .5(80K) (180K) = 84K
EMV(a3) = .2(150K) + .5(90K) ( 40K) = 87K
The optimal act a* = a1, and EMV*= 100K

15. Expected Opportunity Loss (EOL) Decision; Example
Example 23.1 – continued
Use the EOL criterion to find the optimal act for the investment problem presented in this example
EMV(a1) = .2(50K) (0K) (80K) = 34K
EMV(a2) = .2(200K) + .5(20K) (0K) = 50K
EMV(a3) = .2(0K) (10K) (140K) = 47K
The optimal act a* = a1, and EOL* = 34K.

16. Rollback Technique for Decision trees
The probabilities are added to the decision tree on the branches associated with each state of nature
$100,000
A decision node a1
-$50,000
P(s1)=.2 a2
P(s2)=.5
$80,000
P(s3)=.3
$180,000 a3
$150,000
P(s1)=.2
P(s2)=.5
$90,000
$40,000
P(s3)=.3

17. Rollback Technique for Decision trees
Calculate the expected monetary value for all the round nodes, beginning at the end of the tree
$100,000
A decision node a1
(.5)($80,000)
(.3)($180,000)
(.2)(-$50,000)
(+)
(.5)($80,000)
(.3)($180,000)
(.2)(-$50,000)
(+)
(.5)($80,000)
(.3)($180,000)
(.2)(-$50,000)
(+)
(.5)($80,000)
(.3)($180,000)
(.2)(-$50,000)
(+) a2
(.5)($80,000)
(.3)($180,000)
(.2)(-$50,000)
(+)
$84,000 a3
(.2)($150,000)
(.5)($90,000)
$87,000
(+)
(.3)($40,000)

18. Rollback Technique for Decision trees
Select the branch with the most favorable EMV at each square node
$100,000
A decision node a1 a2
$ 84,000 a3
$ 87,000

19. 23.3 Acquiring, Using, and Evaluating Additional Information
Additional information can be acquired and incorporated in the decision process.
Additional information has attendant costs.
The usefulness of the additional information is weighed against its costs.

20. Expected payoff with Perfect Information (EPPI)
EPPI is the maximum price a decision maker should be willing to pay for any information.
States of Nature
Acts a1 a2 a3 s1
100,000
-50,000
150,000 s2
80,000
90,000 s3
180,000
40,000
150,000
If we knew that state of nature s1 were certain to occur,
(perfect information) we would decide to act according to a3.

21. Expected payoff with Perfect Information (EPPI)
EPPI is the maximum price a decision maker should be willing to pay for any information.
States of Nature
Acts a1 a2 a3 s1
100,000
-50,000
150,000 s2
80,000
90,000 s3
180,000
40,000
100,000
If we knew that state of nature s2 were certain to occur,
(perfect information) we would decide to act according to a1.

22. Expected payoff with Perfect Information (EPPI)
EPPI is the maximum price a decision maker should be willing to pay for any information.
If we knew that state of nature s3 were certain to occur,
(perfect information) we would decide to act according to a2.
States of Nature
Acts a1 a2 a3 s1
100,000
-50,000
150,000 s2
80,000
90,000 s3
180,000
40,000
180,000

23. Expected payoff with Perfect Information (EPPI)
Since the future state of nature is uncertain, we can only calculate the expected payoff with perfect information: EPPI = .2(150,000)+.5(100,000)+ .3(180,000)= = $134,000
Since the investor could expect $100,000 (EMV*) without perfect information…

24. Expected Value of Perfect Information (EVPI)
The EVPI is calculated by EVPI = EPPI – EMV*
In our example EVPI = 134,000 – 100,000 = $34,000

25. Decision Making with Additional Information
Purchasing additional information may reduce the amount of uncertainty existing in a given decision problem.
Question: What is the maximum a decision maker should be willing to pay for a given source of additional information?

26. Decision Making with Additional Information
Example 23.1 – continued
The investor can hire a consulting firm (IMC) for a fee of $5,000.
To evaluate IMC’s potential contribution to his decision making capability, the investor receives “likelihood probabilities” from IMC.
The form of the information is P(IMC had predicted a state of nature, given that the state of nature has occurred)

27. Decision Making with Additional Information
Example 23.1 – continued
Likelihood probabilities are conditional probabilities of the form P(Ii | sj), where
Ii = an indicator variable that represents the prediction about the occurrence of si provided by the source of new information.
Ii in our problem:
I1= IMC predicts an interest rate increase (s1)
I2= IMC predicts interest rate stay the same (s2)
I3= IMC predicts an interest rate decrease (s3)

28. Decision Making with Additional Information
Example 23.1 – continued
The likelihood probability table:
I1 (predict s1)
I2 (predict s2)
I3 (predict s3) s1
P(I1|s1)=.60
P(I2|s1)=.30
P(I3|s1)=.10 s2
P(I1|s2)=.10
P(I2|s2)=.80
P(I3|s2)=.10 s3
P(I1|s3)=.10
P(I2|s3)=.20
P(I3|s3)=.70
Interpretation: When s1 actually did occur in the past
IMC correctly predicted s1 60% of the time.

29. How to use the additional Information
It is preferable to incorporate the additional information with the decision maker’s own evaluation.
In our problem: It is preferable to combine the likelihood probabilities with the decision maker’s subjective probabilities.
The mechanism used is Bayes’ law.

30. Prior probabilities and Posterior probabilities
Before IMC is contracted to provide consulting services, the estimated probabilities (prior probabilities) are:
P(s1) = .2; P(s2) = .5; P(s3) = .3
To re-evaluate these probabilities when including the new information we need to find the posterior probabilities:
P(s1|Ii); P(s2|Ii); P(s3|Ii)

31. Finding Posterior Probabilities Using a Probability Tree
Let us find P(sj|I1).
P(s1 and I1)=.2(.6) =.12
P(s2 and I1=.5(.05) =.05
P(s3 and I1=.3(.1) = .03
P(I1)= .20
P(I1|s1)=.60
P(I1|s2)=. 1
P(I1|S3)=.1
P(s1)=.2
P(s2)=.5
P(s3)=.3

32. Finding Posterior Probabilities - a table form
Calculating P(sj|I1) sj
Probabilities
P(sj)
P(I1|sj)
P(sj and I1)
P(sj|I1) s1 .2
.60
.2(.60)=.12
.12/.20=.60 s2 .5
.10
.5(.10)=.05
.05/.20=.25 s3 .3
.3(.10)=.03
.03/.20=.15
P(I1)=.20

33. Finding Posterior Probabilities - a table form
Calculating P(sj|I2) sj
Probabilities
P(sj)
P(I2|sj)
P(sj and I2)
P(sj| I2) s1 .2
.30
.2(.30)=.06
.06/.52=.115 s2 .5
.80
.5(.80)=.40
.40/.52=.770 s3 .3
.20
.3(.20)=.06
P(I1)=.52

34. Finding Posterior Probabilities - a table form
Calculating P(sj|I3) sj
Probabilities
P(sj)
P(I3|sj)
P(sj and I3)
P(sj| I3) s1 .2
.10
.2(.10)=.02
.02/.28=.071 s2 .5
.5(.10)=.05
.05/.28=.179 s3 .3
.70
.3(.70)=.21
.21/.28=.750
P(I1)=.28

35. Using Posterior Probabilities to calculate EMV
The posterior probabilities are revised prior probabilities.
They replace the prior probabilities in the calculation of EMV.
For each possible prediction made by the information source (IMC) we calculate EMV

36. Using Posterior Probabilities to calculate EMV
Finding EMV(ai| I1) from the payoff table and the posterior probabilities P(sj| I1) I1 I1 I1 I1
States of Nature
Acts a1 a2 a3 s1
100,000
-50,000
150,000 s2
80,000
90,000 s3
180,000
40,000
P(sj|I1) .6
.25
.15
EMV(ai|I1) $100,000 $17,000 $118,500

37. Using Posterior Probabilities to calculate EMV
Finding EMV(ai| I2) from the payoff table and the posterior probabilities P(sj| I2)
States of Nature
Acts a1 a2 a3 s1
100,000
-50,000
150,000 s2
80,000
90,000 s3
180,000
40,000
P(sj|I2)
.115
.770
EMV(ai|I2) $100,000 $76,550 $91,150

38. Using Posterior Probabilities to calculate EMV
Finding EMV(ai| I3) from the payoff table and the posterior probabilities P(sj| I3)
States of Nature
Acts a1 a2 a3 s1
100,000
-50,000
150,000 s2
80,000
90,000 s3
180,000
40,000
P(sj|I3)
.071
.179
.750
EMV(ai|I3) $100,000 $145,770 $56,760

39. Making a Decision Summary of the EMV calculations Actions a1 a2 a3 I1
ICM predicts
Actions a1 a2 a3 I1
$100,000
$17,000
$118,500 I2
$76,550
$91,150 I3
$145,770
$56,760
Optimal
Act a3 a1 a2

40. Preposterior Analysis
Should the decision maker pay the price quoted for the additional information?
We evaluate the improvement of the EMV when purchasing the new information.
The added value of the EMV is then compared to the price quoted for the additional information.

41. Expected Value of Sampling Information (EVSI)
116,516
EMV’ =
.20(118,500) +
.52(100,000) +
.28(145,770) = a2 a3
118,500
P(I1)=.20 a1
100,000
P(I2)=.52
Hire IMC a2 a3
Expected Value of Sample Information =
EVSI = EMV’ – EMV* =
– 100,000 =
$16,516
P(I3)=.28 a1 a2
145,770
Do not hire IMC a3 a1
EMV* = 100,000 a2 a3