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Decision trees involving continuous probability distributions 151
However, in some problems the number of possible outcomes may be
very large or even infinite. Consider, for example, the possible percentage
market share a company might achieve after an advertising campaign
or the possible levels of cost which may result from the development of
a new product. Variables like these could be represented by continuous
probability distributions, but how can we incorporate such distributions
into our decision tree format? One obvious solution is to use a discrete
probability distribution as an approximation. For example, we might
approximate a market share distribution with just three outcomes:
high, medium and low. A number of methods for making this sort of
approximation have been suggested, and we will discuss the Extended
Pearson-Tukey (EP-T) approximation here. This was proposed by Keefer
and Bodily, 4 who found it to be a very good approximation to a wide
range of continuous distributions. The method is based on earlier work
by Pearson and Tukey 5 and requires three estimates to be made by the
decision maker:
(i) The value in the distribution which has a 95% chance of being
exceeded. This value is allocated a probability of 0.185.
(ii) The value in the distribution which has a 50% chance of being
exceeded. This value is allocated a probability of 0.63.
(iii) The value in the distribution which has only a 5% chance of being
exceeded. This value is also allocated a probability of 0.185.
To illustrate the method, let us suppose that a marketing manager has
to decide whether to launch a new product and wishes to represent on
a decision tree the possible sales levels which will be achieved in the
first year if the product is launched. To apply the EP-T approximation
to the sales probability distribution we would need to obtain the three
estimates from the decision maker. Suppose that she estimates that there
is a 95% chance that first-year sales will exceed 10 000 units, a 50%
chance that they will exceed 15 000 units and a 5% chance they will
exceed 25 000 units. The resulting decision tree is shown in Figure 6.6(a),
while Figure 6.6(b) illustrates how the discrete distribution has been
used to approximate the continuous distribution.
Of course, in many decision trees the probability distributions will
be dependent. For example, in our product launch example one might
expect second-year sales to be related to the sales which were achieved
in the first year. In this case, questions like the following would need
to be asked to obtain the distribution for second-year sales: ‘Given that
first-year sales were around 25 000 units, what level of sales in the second
year would have a 50% chance of being exceeded?’