A new casino offers the following game: you toss a coin until it comes up heads. If the first heads shows on the N-th toss, you win 2N 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 autility 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 decision-making.

Su, Francis E., et al. “St. Petersburg Paradox.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
Phil Straffin, Game Theory and Strategy.

Fun Fact suggested by:
Francis Su

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