Marginal Utility - Behavioral Mathematics for Game AI - page 179

If we had taken that into consideration as we decided whether to acquire the
Health Kit or the Armor first, our outcome may have been different. The solution
to that problem may have been to assign a utility to the Health Pack that differed
from its “face value.� That is, treat it as less than 50 if we have high health, but more
than 50 if we are running low. That way, when we considered our health status at
the end of each leg, we may have revised our decision as to whether or not to get the
Armor or the Health Kit first. To repeat, if our health is low (e.g., 20), 50 points of
added health mean more to us than adding the same 50 points when our health is
high (e.g., 80).
Increasing Is Not Always the Inverse of Decreasing
It is worth noting that the Health Kit example could have been expressed in terms
of decreasing marginal utility as well. However, caution must be exercised in trans-
posing the utility curve. In Figure 8.6, our level of health is expressed as decreasing
as we move from left to right. Consequently, the utility of the Health Kit is increas-
ing as we moved from left to right as well. On the left side, we express very little util-
ity for the Health Kit, as our health level is high. On the right, as our health nears
zero, the Health Kit is of great utility. When dealing with marginal utility, however,
the shape of the curve is important. It is not enough to simply say “low utility when
health is high; high utility when health is low.� How the marginal utility changes
over the course of the progression is important as well.
Again referring to Figure 8.6, there is no movement in the utility of the Health
Kit when our health is high (the left side). Only at the end (when health gets criti-
cal) do we see major movement in the marginal utility of the Health Kit. The rate
of increase in the utility (the marginal rate) is at its greatest just as our health value
approaches zero. This makes sense to us—the urgency of acquiring a Health Kit gets
significant as we are about to die.
When we reverse the axis so that health is now increasing from left to right, we
want to represent the marginal utility of the Health Kit as decreasing from left to
right (Figure 8.7). However, we must take care to replicate the same progressive
effect on the marginal rate. Both lines in the graph represent decreasing marginal
utility. The difference between them is when that change is at its greatest rate.
If we were to duplicate the effect in Figure 8.6, we would need to use the solid
line in Figure 8.7. This is best examined by working from the right side of the graph
( n ) and back toward zero at the left. Traversing the graph in this direction gives us
the same result as that illustrated in Figure 8.6: The utility for a Health Kit stays
small and then slowly increases until the urgency rises significantly close to zero.