The Value of Information

•The Oil Wildcatter revisited

•Imperfect information

•Revising probabilities

•Bayes’ theorem

The pervasive role of information in decision making is illustrated by the following:

•Should a consumer products firm undertake an expensive test-market program before launching a new and highly promising product?

•What scientific research programs should the government support in the war on cancer?

•What do polls and statistical analysis indicate about the outcome of upcoming senate races?

•How can information on public risks—such as those posed by nuclear power, steel fatigue on bridges or aircraft, or the spread of infectious diseases—be used to prevent disasters.

How can we use information to make better decisions?

The Oil Wildcatter Revisited

Suppose the wildcatter partners with a geologist. For a cost, a seismic test can be performed to obtain better information about drilling

prospects.

We begin with a “perfect” seismic test—that is, a test that gives perfect information as to whether a site is “wet” or “dry.”

Figure 9.1

A Perfect Seismic Test

Note that “good” means oil is present at the site for certain and “bad” means no oil with certainty.

Notes on Figure 9.1How well-off is the partnership with the perfect seismic test?

•Recall that the wildcatter estimated that the probability of finding oil was 0.4—or Pr(W) = 0.4.

•Since good tests occur precisely when the site is wet, then the probability of a good test is also 0.4.

•The probability of a bad test is 0.6—or Pr(B) = 0.6

•A “good” test means the partnership will drill and a “bad” test means no drilling. Therefore the initial expected value is given by:

000,240$)0)(6.0()000,600)(4.0()( vE

Expected Value of Information (EVI)

How valuable is the information provided by the seismic test? To

find out, the compare the expected value of drilling with

information to the expected value without the information.

EVI = Expected value with information

- Expected value without information

Computing EVI

thousand120$)200)(6.0()600)(4.0()( vE

Recall that the expected value of drilling without the information provided by the seismic test was given by:

Thus EVI is given by:

thousand120$120240EVI

A decision maker should acquire costly information if and only if the expected value of the information exceeds its costs.

Do the seismic test if the cost of the test is less than

the EVI.

Imperfect Information

In reality, a seismic test will not give you “perfect” information about whether a site is wet or

dry. You can get better information, however.

This table provides a record of 100 past sites (similar to the current site) where seismic tests have been performed.

•In 30 cases the seismic test indicated “good” and the site was wet.

•In 20 cases the seismic test indicted “good” but the site was dry.

•In 10 cases the seismic test indicated “bad” but the site was wet.

•In 40 cases the seismic test indicated “bad” and the site was dry.

Conditional ProbabilitiesThe results in Table 9.1 allow us to compute conditional probabilities.

G)|WPr(Interpretation: The probability that a site is wet given, or conditional upon, a good seismic test.

Thus:

•Pr(W | G) = 30/50 = 0.6 → meaning, the probability of striking oil given a good seismic test is 0.6.

•Pr(W | B) = 10/50 = 0.2 → meaning, the probability of striking oil given a bad seismic test is 0.2

Before the seismic test, the probability of striking oil is 0.4—this this is the prior probability. After the

test, the partners will revise probabilities based

on the outcome

Notice also that out of 100 sites tested, 50 tested good and 50 tested bad. Thus the probability of a good test is 0.5

Table 9.1—Again

•Notice we could place decimal points to the left of the numbers in the box above—this gives us a slightly different interpretation.

•For example, the upper left hand entry becomes 0.3. 30 percent of sites tested good and were wet.

•We use the notation Pr(W&G) = 0.3 to denote the probability of this joint outcome.

Figure 9.2

An Imperfect Seismic test

A “good” seismic test boosts the chance of striking oil to 0.6. A “bad” seismic test lowers it.

Notes on Figure 9-2

The “contingent” strategy is best—that is , do the seismic test and drill if

it is “good” (expected value =$280,000) and don’t drill if it’s “bad” (expected value = $0). How much do

we gain by using this strategy?

To answer this question, calculate the expected profit at the initial chance node—that is, before the seismic test is performed:

000,140$)0)(5.0()000,280)(5.0()( vE

Recall that the expected profit without the test is $120,000. Thus we can find the EVI:

EVI = $140,000 - $120,000 = $20,000

Thus we would not do the seismic test if it cost more than $20,000

Revising Probabilities

3/4W)|Pr(G

The vendor of the seismic test certifies beforehand that wet sites tested “good” ¾ of

the time. Also, dry sites tested “bad” 2/3 of the time.

Formally, we have:

2/3B)|Pr(D

Remember we assess that 40 percent probability that the site is

wet prior to the seismic test—that is: Pr(W) = 0.4. How can we derive Pr(W | G) and Pr(D | B), the two

critical pieces of information?

Computing Joint Probabilities

Pr(W)W)|Pr(GG)&Pr(W

To derive the joint probability of a wet site and a good test --PR(W&G), we multiply the (conditional) probability of a good test given a wet site—Pr(G | W)—times the (prior) probability of a wet site—Pr(W). That is:

[9.1]

The probability of a given test result—say Pr(G) can be found adding across the appropriate row in Table 9.1

G)&Pr(DG)&Pr(W(G)Pr [9.2]

Calculating Revised Probabilities

Pr(G)

G)&Pr(WG)|WPr(

0.3)(0.75)(0.4G)&(WPr

0.50.20.3G)Pr(

[9.3]

To compute the (conditional) probability of a wet site given a good seismic test:

Thus we have:

6.00.5

0.3G)|WPr(

Thus:The probability of a wet site given a good test is 0.6

We use the same method to compute the (conditional) probability

of a wet site given a bad test.

2.05.0

1.0

4.01.0

4.025.0

B)&Pr(DB)&Pr(W

Pr(W)W)|BPr(

Pr(B)

B)&Pr(WB)|WPr(

The preceding illustrates how we were able to derive

the probabilities for our decision tree with only the

following information: (1) the prior probability of striking

oil—Pr(W); and (2) the vendor’s information about

the reliability of the test.

Check Station 1, p. 358

Suppose the partners face the same seismic test just discussed but are less optimistic about the site; the prior probability is now Pr(W) = 0.28. Construct the joint probability table and compute Pr(W | G) and Pr(W | B).

Bayes’ Theorem

Suppose we have estimated prior probabilities for events we are concerned with, and then obtain new information.

We would like to a sound method to computed revised

or posterior probabilities. Bayes’ theorem gives us a

way to do this.

W)][Pr(Pr(G)

W)|Pr(GG)|WPr(

[9.3]

Bayes’ Theorem: Textbook Definition

W)][Pr(Pr(G)

W)|Pr(GG)|WPr(

[9.3]

This theorem expresses the conditional

probability needed for a decision in terms of the

reverse conditional probability and the prior

probability

Probability Revision using Bayes’ Theorem

PriorProbabilities

NewInformation

Application ofBayes’

Theorem

PosteriorProbabilities

Health Risks from SmokingLC)][Pr(

Pr(S)

LC)|Pr(SS)|LCPr(

We want to derive the probability that an individual will develop lung cancer given the individual is a smoker.

•1 in 12 adults is a heavy smoker.

•The probability that being a smoker given that you are a lung cancer victim is 0.33.

•Thus we have

Pr(LC)4LC)][Pr(8333.0

0.333S)|LCPr(

A smoker is 4 times as likely to develop lung cancer.

A New Seismic Test

Suppose the quality of a new seismic test is summarized in the following table. What is the EVI of this test?

Note that:

•Pr(W | G)= .1/.3 = 0.2; and

•Pr(W | B) = .3/.8 = .375

Computing the EVI with the New Seismic Test

-$200

$600

$600

-$200

Wet

Wet

Dry

Dry

Do not drill

Do not drill

.5

.5

.375

.625

Good test

Bad test

Imperfect

Test

100100

.2

.8

200200

120

Notice that in this case we will drill even if the test is bad—since the expected value of doing so is $100 thousand.

Valueless Information

The information provided by the seismic test isn’t worth

anything because it does not increase expected profit.