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232 Revising judgments in the light of new information
$57000
Do not buy test
Buy test
Plant potatoes
Plant alternative
$62155
$57000
0.41
Virus present
Test indicates virus absent
Virus absent Virus present
0.7
0.3
$30000
Test indicates virus present
0.59
−$20000
$90000
$30000
Plant potatoes
Plant alternative
Plant potatoes
Plant alternative
$17400
Virus absent
$30000
$84500
Virus present
Virus absent
$30000
0.66
0.34
$84500 0.05
Figure 8.11 – Determining the expected value of imperfect information
0.95
−$20000
$90000
−$20000
$90000
We next consider the situation where the test indicates that the virus
has been eliminated. Figure 8.10(b) shows the appropriate calculations.
The probability of the test giving this indication is 0.59 (we knew that it
would be 1 −0.41 anyway) and the posterior probabilities of the presence
and absence of the virus are 0.05 and 0.95, respectively. Again, these
probabilities can now be added to the decision tree.
Let us now determine the expected payoff of buying the test by rolling
back this part of the tree. If the test indicates that the virus is present
then the best decision is to plant the alternative crop and earn $30 000.
However, if the test indicates the absence of the virus, then clearly the
best decision is to plant the potato crop, since the expected payoff of this
course of action is $84 500. This means that if the manager buys the test
there is a 0.41 probability that it will indicate the presence of the virus,
in which case the payoff will be $30 000, and a 0.59 probability that it
will indicate the absence of the virus, in which case the expected payoff
is $84 500. So we have:
the expected payoff with the imperfect information
from the text = 0.41 × $30 00 + 0.59 × 84 500 = $62 155
the expected payoff without the test = $57 000
so the expected value of the imperfect information = $5 155