The Concept of Utility - Behavioral Mathematics for Game AI - page 153

We still have a situation where V ab > V ba , although the margin is a little tighter.
The 210 out of 300 possible points is only 70%. We have lost 5% of the value that
we accomplished in the first example. However, it is still greater than the 65%
that we would achieve by going to B first. The best decision is still goal A, then B.
The fact that the margin of importance is decreasing tips us off to something.
As the relative value of goal B increases as compared to goal A, we are no longer
quite as sure of our decision to take on goal A first. What would happen if this trend
were to continue?
FIGURE 7.17 If the value of goal B is significantly greater than goal A, it becomes
more worthwhile to travel the extra distance to goal B first to accomplish it as soon
as possible before the value decays too much. We can then backtrack to goal A.
Let's try a third example (Figure 7.17). This time, we will increase the value of
goal B to 400. All other values remain the same as before. This means the total value
of all goals is now 500.
This time, V ba > V ab . We have passed a threshold where the importance of goal
B has increased enough to warrant us going out of our way to accomplish it and
only afterward backtracking to touch on goal A. In fact, if we were to put the equa-
tions up against each other and solve for V b , we would find that when V b = 300, the
two solutions are equal… it wouldn't matter in which order you visited them.
The only reason the order was a factor in this decision is because there was a
mechanism in play that caused time to be a factor—namely the decay rate ( r ) of the