Game Development Reference
In-Depth Information
Marginal Utility of Risk
As interesting as the mental exercise of pondering “how much is too much?� can be,
it is still a little difficult to see what is at stake. After all, we can win an
infinite
amount of money. Who wouldn't want that? Why artificially limit it? Well, things
become a little more sobering when the other side of the wager is analyzed. How
much are we willing to
risk
in this lottery? The reason this isn't quite the same
problem is that the scale of the
x-
axis is a little more defined. After all, most of us
don't have an infinite amount of money to plop down on a wager. In fact, we usu-
ally have a maximum net worth with which we can work. And that draws the
proverbial “line in the sand� on making our decisions.
As we discussed earlier, marginal utility can increase. Just as with the curve for
the reward, we can envision a similar curve for risk such as the one on the right side
of Figure 8.8. In this case, the utility change of the initial money is not that big of a
deal for most people. Wagering a dollar is fine for most of us. The second dollar as
well. As the
value
changes early on, the
utility
difference is not all that significant.
However, there's a point where we start to raise our eyebrows a bit. Perhaps not if we
go dollar by dollar… but more likely if, as above, we go $100 at a time. Naturally,
if we start doubling the values, things get out of hand quickly.
Maximizing the Difference
Either way, there comes a point where the respective utilities of the money we are
risking and the money we could win are moving in opposition to one another. Any
additional money we stand to lose means more and more to us, that is, the marginal
utility goes up significantly. On the other hand, the marginal utility of any addi-
tional money we stand to win gets smaller. Eventually, we arrive at a conflict in which
the additional risk is not worth the additional reward.
The answer to the St. Petersburg lottery question—“How much would you be
willing to wager?�—is, therefore, highly subjective when expressed in terms of
value
(which is, after all, what the question is asking). The secret is for each person to find
the “sweet spot� where there is a maximization of the ratio of the
utility
—not
value
—that would be gained over the
utility
of what is being risked.
If we were to overlay the utility curves from Figure 8.8, we would see, in a very
abstract sense, this point illustrated in Figure 8.9. There is a point where the two
curves are the furthest apart. In English, “For
that
kind of payout, I can risk this
much!� That is the maximization point. Accordingly, there is a point where the two
curves cross. Once again, in lay terms, “It's not worth the risk.�
Given the scenario of the St. Petersburg lottery, I wouldn't be willing to wager
more than a dollar. (Perhaps a by-product of running the numbers too much?)
