Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 23.1 Chapter 23 Decision Analysis

23.1Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.

Chapter 23

Decision Analysis


Decision Analysis…

In decision analysis:

• We deal with the problem of selecting one alternative from a list of several possible decisions.

• There may be no statistical data, or if there are data, the decision may depend only partly on them, and

• Profits and losses are directly involved.

We will draw upon concepts from probability theory,

Bayes’ Law, and expected value.


Decision Problem – Terminology…

Choices, decisions, possibilities, alternative courses of action, decision alternatives… these are referred to as acts (ai).

E.g.: a1: Invest in a GIC, or

a2: Invest in the stock market.

Acts are controllable; they reflect our choices.



What actually comes to pass in the future, an outcome, is referred to as a state of nature (si):

E.g.: s1: Interest rates increase, or

s2: Interest rates stay steady or decline.

States of nature are uncontrollable. When enumerating states of nature, we should define all possible outcomes, (i.e.

the list should be mutually exclusive and collectively exhaustive)



For each combination of an act and a state of nature, the amount of profit is calculated. All this data: acts, states of nature, and profits are summarized into a payoff table:

Acts:

States of Nature: a1 a2 … aj

s1 payoff1,1 payoff1,2 … payoff1,j

s2 payoff2,1 payoff2,2 … payoff2,j

: : : : :

si payoffi,1 payoffi,2 … payoffi,j

E.g. This is the payoff we will receive IF we take action a2 AND s1 comes to pass in the future…



An opportunity loss is the difference between

what the decision maker’s profit for an act is

and what the profit could have been

had the best decision been made.

Opportunity loss is calculated row-wise, by taking the combination of act & state of nature with the highest value and then subtracting this maximum value from all the payoffs in the row. If done correctly, we will be left with one zero (where the maximum payoff was located) and positive numbers for all other act / state of nature combinations.


Example 23.1…

Let’s put some numbers to these concepts…

A man wants to invest $1 million for 1 year. He has three possible alternatives (i.e. acts)

a1: Invest in a GIC paying 10%.

a2: Invest in a bond at 8%.

a3: Invest in stocks.

Acts:

a1•GIC a2•Bond a3•Stock


Example 23.1…

Our investor concludes that there are three possible outcomes that can happen during the year his money is invested (i.e. states of nature)

s1: Interest rates increase.

s2: Interest rates stay the same.

s3: Interest rates decrease.

States of Nature:

s1•R>0

s2•R=0

s3•R<0


Example 23.1…

Our investor determines the amount of profit he will make for each possible combination of an act and a state of nature and we create a profit table…

Acts:

States of Nature:


s1•R>0 $100,000 -$50,000 $150,000

s2•R=0 $100,000 $80,000 $90,000

s3•R<0 $100,000 $180,000 $40,000

E.g. We expect to lose $50,000 IF we invest in an 8% bond AND the interest rate goes up in the future…


Example 23.1…Acts:

States of Nature:


s1•R>0 $100,000 -$50,000$150,00

0

s2•R=0 $100,000

$80,000 $90,000

s3•R<0 $100,000$180,00

0$40,000

opportunity loss table Acts:

States of Nature:


s1•R>0 $50,000 $200,000 $0

s2•R=0 $0 $20,000 $10,000

s3•R<0 $80,000 $0 $140,000

Opportunity loss is calculated row-wise, by taking the combination of act & state of nature with the highest value and then subtracting this maximum value from all the payoffs in the row…

If done correctly, we will be left with one zero (where the maximum payoff was located) and positive numbers for all other act / state of nature combinations…


Decision Trees…

Often a decision maker must choose between sequences of acts. Here a payoff table will not suffice to determine the best alternative; instead, we require a decision tree.

Though similar to a probability tree, a decision tree represents acts and states of nature sequentially (chronologically). By convention:

a square node is used to denote a point where a decision (act) is made, and

a circular node represents a state of nature…


Decision Tree…

decision

state of nature

start with a decision node

$100,000*

–$50,000

$80,000

$180,000

$150,000

a1

a2

a3 s1s2

s3

s1s2

s3

$90,000

$40,000 *A GIC (a1) always pays $100,000 regardless of

(si)t


Expected Monetary Value Decision…

Often it is possible to assign probabilities to the states of nature, that is some values P(si).

Our investor believes that future interest rates are most likely to remain essentially the same as they are today and that (of the remaining two states of nature) rates are more likely to decrease than to increase. His probabilities might be:

P(s1) = .2

P(s2) = .5

P(s3) = .3


Expected Monetary Value Decision…

Since we’ve determined probabilities for the states of nature, we can compute the expected monetary value (EMV) of each act…

EMV(aj) = P(s1)xPayoff1,j + P(s2)xPayoff2,j + … + P(si)xPayoffi,j

Hence in our example: EMV(a1) = .2(100,000) + .5(100,000) + .3(100,000) = $100,000

EMV(a2) = .2(-50,000) + .5(80,000) + .3(180.000) = $84,000

EMV(a3) = .2(150,000) + .5(90,000) + .3(40,000) = $87,000

Hence we choose the act (a1) with the highest expected monetary value; i.e. EMV* = $100,000


Expected Monetary Value…

We can use our payoff table for this purpose as well…

EMV* = $100,000

Acts:

States of Nature:


s1•P=.20 $100,000 -$50,000 $150,000

s2•P=.50 $100,000 $80,000 $90,000

s3•P=.30 $100,000 $180,000 $40,000

EMV: $100,000

$84,000 $87,000


Expected Opportunity Loss (EOL) DecisionWe can also calculate the expected opportunity loss (EOL) of each act by using the opportunity loss table and our probabilities for states of nature…

opportunity loss table Acts:

States of Nature:


s1•P=.20 $50,000 $200,000 $0

s2•P=.50 $0 $20,000 $10,000

s3•P=.30 $80,000 $0 $140,000

EOL: $34,000 $50,000 $47,000


Rollback Technique for Decision Trees… Add the probabilities for the states of nature to the branches on our probability tree…

$100,000

–$50,000

$80,000

$180,000

$150,000

a1

a2

a3

P(s1)=.20

$90,000

$40,000

P(s2)=.50

P(s3)=.30

P(s1)=.20

P(s2)=.50

P(s3)=.30

P(s)=1.00


Rollback Technique for Decision Trees… Calculate the expected monetary value (EMV) for each round node…

$100,000

–$50,000

$80,000

$180,000

$150,000

a1

a2

a3

P(s1)=.20

$90,000

$40,000

P(s2)=.50

P(s3)=.30

P(s1)=.20

P(s2)=.50

P(s3)=.30

EMV=(.20)(-50,000) +(.50)(80,000) +(.30)(180,000) =$84,000

P(s)=1.00 EMV=(1)(100,000) = $100,000

EMV=(.20)(150,000) +(.50)(90,000) +(.30)(40,000) =$87,000


Rollback Technique for Decision Trees… Calculate the expected monetary value (EMV) for each round node…

$100,000

–$50,000

$80,000

$180,000

$150,000

a1

a2

a3

$90,000

$40,000

$84,000

$100,000

$87,000


Rollback Technique for Decision Trees… At each square node, we make a decision by choosing the branch with the largest EMV.

$100,000

–$50,000

$80,000

$180,000

$150,000

a1

a2

a3

$90,000

$40,000

$84,000

$100,000

$87,000


Additional Information…

We can acquire useful information from consultants, surveys, or other experiments in an effort to improve our decision process, but this additional information comes at a cost to us.

What’s the maximum price we’d be willing to pay for such a survey?

This leads us to the concept of expected payoff with perfect information (EPPI).


Expected Payoff With Perfect InformationIf we knew with certainty which state of nature would come to pass, we would would make our decisions accordingly.

If our investor knew for sure that state of nature #1(s1 – interest rates rising) were to come to pass, he would invest in stocks in order to earn $150,000 (instead of investing in GICs [return = $100,000] which we calculated when we didn’t have perfect information).

Back to our payoff table – what is the highest payoff for each state of nature (each row)?


Expected Payoff With Perfect InformationWe calculate our Expected Payoff With Perfect Information (EPPI) as any other expected value (i.e. use P(si)’s)

EPPI = .2(150,000) + .5(100,000) + .3(180,000) = $134,000

Acts:

States of Nature:


s1•P=.20 $100,000 -$50,000$150,00

0

s2•P=.50 $100,000

$80,000 $90,000

s3•P=.30 $100,000$180,00

0$40,000


Expected Value of Perfect Information…Without perfect information, our investor could put his money into a GIC and earn EMV* = $100,000 expected profit.

Subtracting EMV* from EPPI leaves us with a quantity known as EVPI: the Expected Value of Perfect Information

EVPI = EPPI – EMV* = $134,000 – $100,000 = $34,000


Expected Value of Perfect Information…Expected Value of Perfect Information

EVPI = EPPI – EMV* = $134,000 – $100,000 = $34,000

The expected value of perfect information is $34,000 more than what our investor could earn without perfect information.

Thus EVPI provides an upper bound for survey costs to improve the decision making information on hand.


Decision Making w/ Additional InformationSuppose our investor wants to improve his decision-making capabilities by hiring a consultant. IMC (the consultant), will, for a $5,000 fee, analyze the economic conditions and forecast the behavior of interest rates over the next 12 months.

IMC details of their past successes forecasting interest rates. These are stated as conditional probabilities (also known as likelihood probabilities).

Should we pay the $5,000 fee? Is it worth it to buy this data?


Likelihood Probabilities…

I1: IMC predicts the interest rates will increase

I2: IMC predicts the interest rates will stay the same

I3: IMC predicts the interest rates will decrease

I1(predicts s1)

I2(predicts s2)

I3(predicts 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|s2)=.70

E.g. when interest rates rose in the past, IMC correctly predicted this happening 60% of the time…


Likelihood Probabilities…

P(I1|s1) = .60

When the interest rates actually increased (s1) in the past, IMC correctly predicted this happening 60% of the time.

That means 40% of the time they were wrong.

30% of the time they predicted stable rates and 10% of the time they predicted falling rates, hence the other likelihood probabilities: P(I3|s1) = .10 (say)

We refer to Ii as an “indicator variable”.


Terminology…

Our original probabilities, P(si) are called prior probabilities — they were determined prior to any new information.

We have likelihood probabilities, P(Ii | sj) from the company providing the information.

What we’re interested in finding are posterior probabilities (a.k.a. revised probabilities), which are of the form P(sj | Ii) – what is the probability of a state of nature occurring, given that it was indicated by additional information.

That is: what’s the probability the consultant will be right!


Posterior Probabilities…

We can compute posterior probabilities for each indicator variable in turn with this approach:

P(sj) – prior probabilities; known.

P(Ii | sj) – likelihood probabilities; known.

P(sj and Ii) = P(sj) x P(Ii | sj) – from:

P(Ii) = sum of all P(sj and Ii)’s

P(sj | Ii) = P(sj and Ii) / P(Ii) – again, from

Thus we have our posterior probability! (You may recall this as Bayes’ Law from earlier)


Calculating Posterior Probabilities…(I1) and are given, we use to get

sj

P(sj)

P(I1|sj)

P(sj and I1)

s1 .20 .60(.20)

(.60)=.12

s2 .50 .10(.50)

(.10)=.05

s3 .30 .10(.30)

(.10)=.03


Calculating Posterior Probabilities…(I1)Add up column to get the marginal probability for I1 –

sj

P(sj)

P(I1|sj)

P(sj and I1)

s1 .20 .60(.20)

(.60)=.12

s2 .50 .10(.50)

(.10)=.05

s3 .30 .10(.30)

(.10)=.03

P(I1)=.20


Calculating Posterior Probabilities…(I1)Again, using divide the values in by to get

sj

P(sj)

P(I1|sj)

P(sj and I1)

P(sj|I1)

s1 .20 .60(.20)

(.60)=.12.12/.20=.60

s2 .50 .10(.50)

(.10)=.05.05/.20=.25

s3 .30 .10(.30)

(.10)=.03.03/.30=.15

P(I1)=.20

*repeat this process for indicators I2 and I3…


Posterior Probabilities…

After the probabilities have been revised (i.e. made into posterior probabilities) we can use them in exactly the same way we used the prior probabilities — to calculate the expected monetary value of each act:

EMV(a1) = .60(100,000) + .25(100,000) + .15(100,000)= $100,000

EMV(a2) = .60(-50,000) + .25(80,000) + .15(180,000)= $17,000

EMV(a3) = .60(150,000) + .25(90,000) + .15(40,000)= $118,500

Thus, if IMC forecasts an increase in interest rates, the best course of action is act a3 (since it has the highest EMV).

*repeat this process for indicators I2 and I3…


Summary of the EVM Calculations…

The action with the highest expected monetary value will change depending on what the consultant predicts…

Acts:

ICM Predicts


I1 $100,000 $17,000$118,50

0

I2$100,00

0$76,550 $91,150

I3 $100,000$145,77

0$56,760


Preposterior Analysis…

But we still haven’t answered the question – is it worth $5,000 to buy the consultant’s information or go it alone?

If IMC predicts s1, choose a3, payoff = $118,500



We also know the marginal probabilities…

P(I1) = .20, P(I2) = .52, P(I3) = .28



We can calculate an expected monetary value with additional information as the weighted average of:

the expected monetary values, and

P(I1), P(I2), and P(I3) as weights.

Hence:EMV' =.20(118,500) + .52(100,000) + .28(145,770)= $116,516



The value of the consultant’s forecast is the delta between

the expected monetary value with additional information (EMV')

and the expected monetary value without additional information (EMV*).

This difference is called the expected value of sample information and is denoted EVSI. Thus,

EVSI = EMV' - EMV* = $116,516 - $100,000 = $16,516


Expected Value of Sample InformationSince the expected value of sample information is greater than the cost of obtaining the information…

$16,516 > $5,000

…we should advise our investor to hire IMC consultants and obtain an interest rate forecast before investing.

Documents

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 23.1 Chapter 23 Decision Analysis