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Assessing the value of new information 225
Profit $20 000 $80 000 $140 000 $220 000
Utility 0 0.5 0.8 1.0
The retailer estimates that there is a 0.4 probability that sales will be low
and a 0.6 probability that they will be high. What level of stocks should
he hold?
A decision tree for the retailer’s problem is shown in Figure 8.7(a). It
can be seen that his expected utility is maximized if he decides to hold a
small stock of the commodity.
Before implementing his decision the retailer receives a sales forecast
which suggests that sales will be high. In the past when sales turned
out to be high the forecast had correctly predicted high sales on 75%
of occasions. However, in seasons when sales turned out to be low the
forecast had wrongly predicted high sales on 20% of occasions. The
underlying market conditions are thought to be stable enough for these
results to provide an accurate guide to the reliability of the latest forecast.
Should the retailer change his mind in the light of the forecast?
We can use Bayes’ theorem to modify the retailer’s prior probabilities
in the light of the new information. Figure 8.7(b) shows the probability
tree and the appropriate calculations. It can be seen that the posterior
probabilities of low and high sales are 0.15 and 0.85, respectively. These
probabilities replace the prior probabilities in the decision tree, as shown
in Figure 8.7(c). It can be seen that holding a large stock would now lead
to the highest expected utility, so the retailer should change his mind in
the light of the sales forecast.
Assessing the value of new information
New information can remove or reduce the uncertainty involved in a
decision and thereby increase the expected payoff. For example, if the
retailer in the previous section was, by some means, able to obtain
perfectly accurate information about the summer demand then he could
ensure that his stock levels exactly matched the level of demand. This
would clearly lead to an increase in his expected profit. However, in
many circumstances it may be expensive to obtain information since it
might involve, for example, the use of scientific tests, the engagement of
the services of a consultant or the need to carry out a market research
survey. If this is the case, then the question arises as to whether it
is worth obtaining the information in the first place or, if there are