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192 Applying simulation to decision problems
choose. The problem is, unlike the examples which we encountered in
Chapter 5, each option has a very large number of possible outcomes.
One way around this problem is to find a mathematical function which
will approximate the decision maker’s utility function. A computer can
then be used to calculate the utility for each profit generated in the
simulation run. The resulting utilities would then be averaged to give
the expected utility.
Stochastic dominance
Sometimes the alternative with the highest expected utility can be
identified by a short cut method which is based on a concept known as
stochastic dominance. This exists where the expected utility of one option
is greater than that of another for an entire class of utility functions. This
means that we can be sure that the option will be preferred without
going to the trouble of eliciting the decision maker’s complete utility
function; all we need to establish is that the utility function has some
basic characteristics. 3
Stochastic dominance can be recognized by plotting the cumulative
probability distribution functions (CDFs). As we saw in Chapter 4, the
CDF shows the probability that a variable will have a value less than
any given value. First- and second-degree stochastic dominance are the
two most useful forms which the CDFs can reveal.
First-degree stochastic dominance
This concept requires some very unrestrictive assumptions about the
nature of the decision maker’s utility function. When money is the
attribute under consideration, the main assumption is simply that higher
monetary values have a higher utility. To illustrate the application of
first-degree stochastic dominance, consider the following simulation
results which relate to the profits of two potential products, P and Q:
Profit
($m) Prob.
Product P Product Q
Cumulative
prob.
Profit
($m) Prob.
Cumulative
prob.
0tounder5 0.2 0.2 0tounder5 0 0
5 to under 10 0.3 0.5 5 to under 10 0.1 0.1
10 to under 15 0.4 0.9 10 to under 15 0.5 0.6
15 to under 20 0.1 1.0 15 to under 20 0.3 0.9
20 to under 25 0.1 1.0