Game Development Reference
In-Depth Information
This is a bit more in line with what we originally intended. The marginal util-
ity of the second and third soldiers is still high, that is, it is still important for us to
build them. Additionally, as we look toward the 9th and 10th soldiers, we see that
the values are much lower than those of the initial soldiers. We have also solved our
problem of negative utility. While the marginal utility continues to decrease, it
approaches but never reaches zero. Even the 100th soldier has
some
marginal util-
ity to us, albeit a very small one.
The difference between the three formulas is apparent when we graph them to-
gether (Figure 8.5). In the case of the first, linear formula (the solid line), the mar-
ginal utility decreased but was destined to cross into negative territory shortly after
the 10th soldier. The dashed line shows the undesirable rapid drop-off in marginal
utility that the second formula provided.
FIGURE 8.5
All three of the example formulas show decreasing
marginal utility yet exhibit very different characteristics such as
the rate of decrease or eventually intercepting the
x
-axis.
The dotted line is the result of the third formula. Its curve is in the same neigh-
borhood as the original, linear formula. Even by simply eyeballing the graph, we
can tell there is a different characteristic, however. It approaches the axis but at a
steadily decreasing rate. As we saw from the data, even if we extended this line out
to 100, it would not have reached a marginal utility of zero. Constructing curves
that exhibit this characteristic is a key to building suitable algorithms in behavioral
mathematics. We will get into more detail on how to approach these issues later in
the topic.
