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Applying simulation to a decision problem 193
Cumulative probability
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Product P
Product Q
0
0 5 10 15 20 25
Profit ($millions)
Figure 7.6 – First-degree stochastic dominance
The CDFs for the two products are plotted in Figure 7.6. It can be
seen the CDF for product Q is always to the right of that for product P.
This means that for any level of profit, Q offers the smallest probability
of falling below that profit. For example, Q has only a 0.1 probability of
yielding a profit of less than $10 million while there is a 0.5 probability
that P’s profit will fall below this level. Because Q’s CDF is always to the
right of P’s, we can say that Q exhibits first-degree stochastic dominance
over P. Thus, as long as the weak assumptions required by first-degree
stochastic dominance apply, we can infer that product Q has the highest
expected utility.
Second-degree stochastic dominance
When the CDFs for the options intersect each other at least once it may
still be possible to identify the preferred option if, in addition to the
weak assumptions we made for first-degree stochastic dominance, we
can also assume that the decision maker is risk averse (i.e. his utility
function is concave) for the range of values under consideration. If
this assumption is appropriate then we can make use of second-degree
stochastic dominance. To demonstrate the application of this concept let
us compare the following simulation results which have been generated
for two other potential products, R and S: