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194 Applying simulation to decision problems
Product R Product S
Profit
($m) Prob.
Cumulative
prob.
Profit
($m) Prob.
Cumulative
prob.
0 to under 5 0.1 0.1 0 to under 5 0.3 0.3
5 to under 10 0.3 0.4 5 to under 10 0.3 0.6
10 to under 15 0.4 0.8 10 to under 15 0.2 0.8
15 to under 20 0.2 1.0 15 to under 20 0.1 0.9
20 to under 25 0 1.0 20 to under 25 0.1 1.0
The CDFs for the two products are shown in Figure 7.7. It can be seen
that for profits between $0 and $15 million, R is the dominant product,
while for the range $15–25 million, S dominates. To determine which is
the dominant product overall we need to compare both the lengths of
the ranges for which the products are dominant and the extent to which
they are dominant within these ranges (i.e. the extent to which one
curve falls below the other). This comparison can be made by comparing
area X, which shows the extent to which R dominates S, with area Y,
the extent to which S dominates R. Since area X is larger, we can say
that product R has second-degree stochastic dominance over product S.
Again, as long as our limited assumptions about the form of the decision
maker’s utility function are correct then we can conclude that R has a
higher expected utility than S.
Cumulative probability
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Area X
Product S
Product R
Area Y
0
0 5 10 15 20 25
Profits ($m)
Figure 7.7 – Second-degree stochastic dominance