The St. Petersburg Paradox

The paradox

A casino offers you a simple game. A fair coin is flipped repeatedly until it lands heads. If heads appears on the first flip, you win $2. If it takes until the second flip, you win $4. If the third, $8. The payout on flip n is $2^n.

The expected value of this game is the sum of each outcome multiplied by its probability: (1/2)×$2 + (1/4)×$4 + (1/8)×$8 + ... Every term equals $1. There are infinitely many terms. The expected monetary value is infinite.

And yet almost no one would pay more than $20 or $30 to play. Many people wouldn't pay $10. Something is wrong with expected monetary value as a theory of rational choice.

Bernoulli's fix: diminishing marginal utility

Daniel Bernoulli's solution was to observe that money does not scale linearly with wellbeing. Going from $0 to $1,000 matters more to a poor person than going from $1,000,000 to $1,001,000. The psychological benefit of money diminishes as you have more of it.

He proposed that rational agents should maximize expected utility, not expected money, and that utility grows as the logarithm of wealth. Under logarithmic utility, the St. Petersburg game has a finite expected value, and the price people are willing to pay matches their actual behavior much better.

This insight launched expected utility theory, which dominated economics for two centuries and still underpins most formal models of rational choice under risk.

Why the problem keeps coming back

Bernoulli's solution works for the original game but doesn't close the paradox. You can construct a modified game where payouts grow fast enough to overcome any proposed utility function, producing the same infinite-expected-value problem at the level of utility rather than money. If your utility function is unbounded, a St. Petersburg-style paradox can always be rebuilt.

The deeper issue is that the paradox is a test case for expected value theories generally. Any framework that assigns finite value to everything and multiplies by probability will break down when payoffs grow without bound. The Pasadena Game takes this further, constructing a game where the expected value doesn't converge at all, not to infinity but to no definite value.

What the St. Petersburg Paradox reveals is that expected value, intuitive and powerful as it is, is not a complete theory of rational choice. The cases where it breaks down are not just mathematical curiosities. They mark the edge of the framework.