The Value of Information

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The Value of Information

• The Oil Wildcatter revisited
• Imperfect information
• Revising probabilities
• Bayes theorem

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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 riskssuch as those
  posed by nuclear power, steel fatigue on bridges
  or aircraft, or the spread of infectious
  diseasesbe used to prevent disasters.

3
How can we use information to make better
decisions?
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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 testthat
is, a test that gives perfect information as to
whether a site is wet or dry.
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A Perfect Seismic Test
Figure 9.1
Note that good means oil is present at the site
for certain and bad means no oil with certainty.
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Notes on Figure 9.1

• How well-off is the partnership with the perfect
  seismic test?
• Recall that the wildcatter estimated that the
  probability of finding oil was 0.4or 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 0.6or 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

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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
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Computing EVI
Recall that the expected value of drilling
without the information provided by the seismic
test was given by
Thus EVI is given by
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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.
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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.
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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.

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

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Before the seismic test, the probability of
striking oil is 0.4this 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
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Table 9.1Again

• Notice we could place decimal points to the left
  of the numbers in the box abovethis 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(WG) 0.3 to denote the
  probability of this joint outcome.

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An Imperfect Seismic test
Figure 9.2
A good seismic test boosts the chance of
striking oil to 0.6. A bad seismic test lowers
it.
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Notes on Figure 9-2
The contingent strategy is bestthat is , do
the seismic test and drill if it is good
(expected value 280,000) and dont drill if
its 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 nodethat is, before
the seismic test is performed
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
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Revising Probabilities
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
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Remember we assess that 40 percent probability
that the site is wet prior to the seismic
testthat is Pr(W) 0.4. How can we derive Pr(W
G) and Pr(D B), the two critical pieces of
information?
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Computing Joint Probabilities
To derive the joint probability of a wet site
and a good test --PR(WG), we multiply the
(conditional) probability of a good test given a
wet sitePr(G W)times the (prior) probability
of a wet sitePr(W). That is
9.1
The probability of a given test resultsay Pr(G)
can be found adding across the appropriate row in
Table 9.1
9.2
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Calculating Revised Probabilities
To compute the (conditional) probability of a wet
site given a good seismic test
9.3
Thus we have
Thus
The probability of a wet site given a good test
is 0.6
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We use the same method to compute the
(conditional) probability of a wet site given a
bad test.
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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 oilPr(W) and (2)
the vendors information about the reliability of
the test.
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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).
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Bayes Theorem
9.3
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.
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Bayes Theorem Textbook Definition
9.3
This theorem expresses the conditional
probability needed for a decision in terms of the
reverse conditional probability and the prior
probability
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Probability Revision using Bayes Theorem
Prior Probabilities
New Information
Application of Bayes Theorem
Posterior Probabilities
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Health Risks from Smoking

• 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

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

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Computing the EVI with the New Seismic Test
Do not drill
Wet
600
.5
Good test
200
200
Dry
.2
-200
Imperfect
Do not drill
.5
120
Wet
Test
600
Bad test
.375
100
100
.8
Dry
-200
.625
Notice that in this case we will drill even if
the test is badsince the expected value of doing
so is 100 thousand.
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Valueless Information
The information provided by the seismic test
isnt worth anything because it does not increase
expected profit.