Game Development Reference
In-Depth Information
hypothetical brick-to-be that we have been using in the example.) For that matter,
maybe the multimillionaire should just buy the warranty and be done with it. After
all, the cost of the warranty is loose change to him.
It all comes down to how important that money is for a person—not so much in
an absolute sense (“there is enough money in my account to cover the warranty…�),
but more so in a relative one (“…but ensuring a working computer in the future is
not
as important
to me as having beer money right now�). What we have witnessed
is the application of marginal utility to something that is being given up (or even
potentially given up). This is the marginal utility of risk.
The St. Petersburg Paradox
The phenomenon of marginal utility of risk was illustrated in spectacular, yet
controversial, fashion by Swiss mathematician, Daniel Bernoulli, in 1738. In intro-
ducing the St. Petersburg paradox, Daniel showed that using a purely mathematical
approach of probability theory to determine decision theory can cause problems.
(Credit where credit's due: The original problem was conceived by Daniel's cousin,
Nicolas Bernoulli. Daniel just presented it and got it published in
Commentaries of
the Imperial Academy of Science of Saint Petersburg
.)
The St. Petersburg paradox is based on a lottery concept of a very simple sort.
The pot starts at one dollar. You then repeatedly flip a coin. On any given round,
if the coin shows tails, you win the pot. If it shows heads, you get to flip the coin
again. Therefore, if you flip tails on the first toss, you would win the initial value of
the pot: $1. However, if you were to flip heads first and your
second
flip showed
tails, you would win $2. If you showed heads on the second flip and tails on the
third, you would win $4. Skipping ahead a bit, if you got on a roll and threw ten
heads in a row before tossing tails, you would win $1,024 (2
10)
. The big question is,
“How much would you be willing to pay to enter this lottery?�
The question makes for an interesting twist on Pascal's charge that we need to
determine what is at stake. In this case, we are going to try to determine what it is
that we want to
put
at stake. The temptation is to look for the purely mathematical
solution to this question. After all, that approach worked admirably for us in deter-
mining how much we should pay for a warranty on our (less than stellar) com-
puter.
The starting point involves trying to figure out how much we are likely to win.
This part, at least, is relatively straightforward mathematically. All of the informa-
tion we need was given to us in the rules of the game. A single flip of a coin is a
50/50 proposition. The pot doubles with each “successful� flip of a head. Let's bust
out the math.
