Game Development Reference
In-Depth Information
T
HE
L
OGIT
F
UNCTION
Close kin to the logistic function is the
logit function
(pronounced
low-jit
). We can
think of a logit curve as a logistic curve rotated 90 degrees. Where the logistic curve
approached
y
= 0 and
y
= 1, the s-curve defined by the logit function approaches
the asymptotes of
x
= 0 and
x
= 1. The formula for the logit function is
As with the logistic function above, we can define any base to the logarithm that
we wanted, although our good friend
e
is typically used (Figure 10.8). By increas-
ing the value of the logarithm base, we can “flatten� the s-curve horizontally.
FIGURE 10.8
We can think of the logit function as a rotated version of the
logistic function. The curve approaches but does not reach
x
= 0 and
x
= 1.
Therefore, we must take care not to attempt to solve for those values of
x
.
As with the logistic function, we can use the logit function in numerous appli-
cations with regard to behavioral and psychological perceptions and reaction. For
example, rather than resembling the notion of
decreasing
marginal utility like the
logistic function, as we move from the “center� point (
x
= 0.5), the logit curve
resembles
increasing
marginal utility.
Because the value of
y
approaches infinity the closer
x
gets to 0 and 1, there
truly are no upper and lower bounds. However, as you can see from Figure 10.8, the
