Marginal Utility - Behavioral Mathematics for Game AI - page 185

many psychological studies have shown, there is a sort of leveling out point. Once
a certain threshold is reached (although that threshold is fairly vague), extra money
isn't as important as it was earlier on in the acquisition process.
In similar fashion to the caffeine I ingest so ravenously in the morning, after a
certain point, people tend to say, “I have enough� with other things as well. Let's say
you have reached approximately $2 million in winnings from your coin-flipping
adventure in the St. Petersburg lottery. Is 4 million really twice as good as only 2
million? What about 8 million? Is that really twice as good as having only four mil-
lion? If you were sitting on $150 million, is your life going to be twice as good if you
double it to $300 million?
The quandary would be a little more obvious if we were working with a fixed
value of money rather than doubling each time. For instance, increasing from $100
to $200 is nice. Increasing from $1,000 to $1,100 isn't as big of a deal. Increasing
from $1,000,000 to $1,000,100 is loose change. Certainly, the effect is more obvious
with this example—particularly because we can see the percentage change that each
additional $100 has. Is it doubling the first hundred? Is it a 10% addition to $1,000?
Or is it a miniscule 0.01% increase when tacked on to a million? However, while it
is more obvious when expressed this way, even doubling the value each time trans-
lates into a diminishing utility at some point. It's simply a matter of scale.
Regardless of scale, we can represent the concept of diminishing marginal util-
ity with a curve such as the one on the left side of Figure 8.8. Note that, despite
being the same width as the areas represented by a , margins b and c show signifi-
cantly less of a vertical increase. In other words, despite the fact that the face value
of a prize in a lottery increases, each additional amount of increase begins to mean
less and less in terms of the utility that we would have for that prize.
FIGURE 8.8 As the value of a prize goes up, the marginal utility of each increase gets
smaller. As the value of a wager goes up, the marginal utility of each increase gets larger.