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Buying perfect information
As manager of a post office, you are trying to decide whether to rearrange a production line and facilities in
order to save labor and related costs. Assume that the only alternatives are to "do nothing" or "rearrange."
Assume also that the choice criterion is that the expected savings from rearrangement must equal or exceed
$11,000.
Based on your experience and currently available information, you predict:
a. The "do nothing" alternative will have operating costs of $200,000.
b. The "rearrange" alternative will have operating costs of $100,000 if it is a success or $260,000 if it is a
failure. You think there is a 60% chance of success and a 40% chance of failure.
Required:
1. Compute the expected value of the costs for each alternative. Compute the difference in expected costs.
2. You can hire a consultant, Joan Zenoff, to study the situation. She would then render flawless prediction
of whether the rearrangement would succeed or fail. Compute the maximum amount you would be willing to
pay for the errorless prediction.
1. Expected value of each alternative:
Do nothing:
Rearrange:
Difference:
2. Value of perfect information:
EVPI = Expected value of the decision with perfect information less the expected value of the decision
without information.
Buying imperfect information.
Refer to the preceding problem. Zenoff's eventual prediction of success or failure will be imperfect. The
following analysis is provided:
EVENTS
PROBABILITY OF EVENT
If optimistic report
If pessimistic report
Success
.818
.333
Failure
.182
.667
Required:
1. Compute the expected costs, assuming an optimistic report.
2. Compute the expected costs, assuming a pessimistic report.
3. The probability of getting an optimistic report is .55; a pessimistic report, .45. The computation of these
probabilities is not necessary for solving this problem. Compute the expected value of imperfect information.
Compare it with the expected value of existing information as computed in the preceding problem. The
consultant fee is $1,000. Should she be hired?
4. Suppose you hire the consultant. The report is pessimistic. Should you proceed with the rearrangement?
Explain.
1. E(Optimistic)
2. E(Pessimistic) =
3. E(Imperfect Info.) =
4.
Probability Revision
Computing the posterior probabilities for the previous example.
1. The consultant is correct in her report (optimistic or pessimistic) 75% of the time.
Let
O = optimistic report
P = pessimistic report
S = rearrangement succeeds
F = rearrangement fails
P(O|S) = P(P|F) = the probability of being correct = .75
P(O|F) = P(P|S) = the probability of being wrong = .25
These are often referred to as likelihoods. The chronological order of events seems wrong.
The marginal probability of obtaining a pessimistic report
= P(P, S) + P(P, F) = P(S)P(P|S) + P(F)P(P|F)
= the probability the rearrangement will succeed times the probability the consultant erroneously predicts a
failure (.6 x .25)
+ the probability the rearrangement will fail times the probability the consultant predicts a failure (.4 x .75)
= (.6 x .25) + (.4 x .75)
= .45.
The probability of an optimistic report
= P(O, S) + P(O, F) = P(S)P(O|S) + P(F)P(O|F)
= the probability of a success times the probability the consultant successfully predicts a success (.6 x .75)
+ the probability the proposal will fail times the probability the consultant erroneously predicts a success (.4 x
.25) giving (.6 x .75) + (.4 x .25)
= .55.
2. The probability of success given an optimistic report is:
P(S|O) =
P(S)P(O | S) P(S, O) .45
=
= .8182, so the probability of failure is

.55
P(O)
P(O)
P(F|O) = (1 - P(S|O)) = .1818
The probability of success given a pessimistic report is:
P(S|P) =
P( S ) P( P | S ) P( S ,P) .15
= .45 = .3333, so the probability of failure is

P( P)
P( P)
P(F|P) = (1 - P(S|P)) = .6667
Note they sum to one.
Review Definitions:
prior probability = probability of state of nature in general with no additional information.
In general, in March, any day, any year, there is P(snow) = .2 and P(no snow) = .8.
likelihood = P(a particular kind of information preceeded a given event)
If the weather turned out to be snowy (event), then there was a 75% chance that the preceding forecast
predicted snow. P(no predicted|snow occurred).
This is not in the chronological order in which I saw the weather (second) and forecast (first).
If I experience a successful rearrangement, there is a conditional probability that the report was optimistic of
P(O|S) =
P( S | O) P( S ) .45
= .60 = .75, the probability of a correct report.
P( S )
posterior probability = P(If information, then subsequent actual event) If the forecast predicts snow, then
there is a x probability it really will snow later. This is the correct chronological order and these probabilities
belong in a decision tree with imperfect information.
"information" = Examples: market research, customer survey, trial run, movie preview, expert opinion, an
audit, financial statements.
expected value of perfect information (EVPI): expected value with perfection - expected value with no
information.
Hence, you know exactly which action to take.
expected value of imperfect information (EVII): expected value with imperfect information - expected value
with no information.
Hence, you will still make some mistakes in choosing the best action, and you must compare the EV(imperfect
information) to its cost to decide whether to use it.