Game Development Reference
In-Depth Information
Translated into English, this reads: The estimated payoff on
n
tosses (as
n
goes
to infinity) is an infinite accumulation of 50 cents, which, of course, is an infinite
payoff. Following our premise that we are willing to wager what we are likely to win
over time, we should be willing to put up an infinite amount of money for the priv-
ilege of playing the St. Petersburg lottery. But is that necessarily feasible?
On one hand, who would be willing to put up an infinite amount of money to
play this lottery? Certainly, none of us has an infinite amount of money, so we
should be a little more reasonable and suggest that we could put up everything we
own for this lottery. Doesn't this sound a little silly? It doesn't matter how many
times I tell you that you could win an infinite amount of money, or even that it is
possible that you could at least win all your assets back. You are still probably not
going to take me up on this deal.
On the other hand, if there is a potential for such a massive payoff, who would
offer this lottery in the first place? Even if the players are wagering massive sums of
money, there is always the possibility that you are going to pay out far more than
what has come in. Jokes of governmental debt aside, I can't even see a city, state, or
country being able to back the potential “infinite� payoff.
So what went wrong? Why won't people be willing to put up massive amounts
of money to play the St. Petersburg lottery? Why won't anyone offer it anyway? The
problem came into play when the nasty little concept of infinity got involved.
The trouble you can get into when infinity is involved is actually worthy of its own
book. (In fact,
To Infinity and Beyond
by Eli Maor is a fun read about the cultural
effect that the notion of infinity has had over time.) However, the lesson we have
learned actually has gone beyond the pitfalls of limitless numbers.
By using infinity, Bernoulli unwittingly crafted a problem that can be posed to
any person, regardless of their personal wealth. The hidden factor that makes this
possible is the underlying question of “how much is this worth to you?� That is a
question that each of us would be forced to ask ourselves as we pondered how
much we would be willing to put on the line in the St. Petersburg lottery. The com-
plicating factor is that there is not
one
factor to consider (and one corresponding
question to ask). There are actually
two
. Both of them involve marginal utility. In
fact, they are actually almost reciprocals of each other.
Marginal Utility of Reward
First, we must consider what value we would put on the money won. Sure, we would
like to win a few bucks. That's always nice. However, as the money goes up, it starts
to look the same. The increase from $1 to $2 on a wager is rather substantial. We
have doubled our money! In fact, the increase from $2 to $4 is just as attractive.
Again, we have doubled our money. The case can be made that having $4 is twice
as good as having $2, just as having $2 is twice as good as having $1. However, as
