A new casino offers the following game: you
toss a coin until it comes up heads. If the first heads
shows on the Nth toss, you win 2^{N} dollars.
(Thus the payoff doubles with each coin toss that isn't
heads.)
How much should you be willing to pay to enter this game?
At first glance, you might think it is the expected
value of the payoff to the player. But if you
calculate it, you get a divergent series...
the expected value is infinite!
If so, then maybe you should be willing to pay any
fixed finite amount of money to play this game?
And yet the chance of winning more than 4 dollars
is only 1/4, so that can't be right, can it?
Presentation Suggestions:
Alternately, you might ask students, if they were the
casino owner, how much they should be charge to play
this game? Or, would they even offer the game?
The Math Behind the Fact:
The more general question is: how should players
evaluate their preferences over options that
involve chance? (Classic examples are games,
but such options could occur in
any decision that you face in your life.)
Naively, you might think that you
should choose the option that gives you the highest
expected value.
The lesson of this paradox is that people (like yourself)
do not play games as if they are maximizing the expected
monetary value they receive. However,
certain rationality assumptions about the way people
behave (the von Neumann and Morganstern axioms)
do imply that people do act as though they are
maximizing something, which is often called a
utility function. See Deal or no Deal for an idea of how to construct a utility function.
This subject is often treated in detail in a course on game theory,
which is the mathematical modeling of decisionmaking.
How to Cite this Page:
Su, Francis E., et al. "St. Petersburg Paradox."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
