The St. Petersburg Paradox: Infinite Expected Value

In 1738, Swiss mathematician Daniel Bernoulli described a gambling game that exposed a fundamental problem with expected value as a guide to decision-making. The game is simple. The implications are profound.

The game

A fair coin is flipped repeatedly until it lands tails. If tails appears on the first flip, you win $2. If tails appears on the second flip, you win $4. Third flip: $8. Fourth: $16. The prize doubles with each flip. How much should a rational person pay to play?

E[X] = (1/2)×$2 + (1/4)×$4 + (1/8)×$8 + ...
      = $1   + $1   + $1   + ...
      = ∞

The expected value is infinite.

The paradox

Expected value theory says you should pay any finite amount to play a game with infinite expected value. Yet when people are asked how much they would pay, most answer between $2 and $25. Almost no one would pay $100. No one would pay $1,000. The theory says any price is rational. Reality says otherwise.

Bernoulli's solution: utility

Bernoulli proposed that people do not maximise expected value — they maximise expected utility. The utility of money is not linear. An extra $1,000 matters much more to someone with $100 than to someone with $1,000,000. Bernoulli suggested logarithmic utility: the value of wealth is proportional to its logarithm.

E[log(wealth)] = Σ (1/2ⁿ) × log(2ⁿ)

This series converges to a finite number,
resolving the paradox.

Practical implications

The St. Petersburg paradox has practical consequences for how we think about risk. It is why insurance exists — people pay more than the expected loss to eliminate variance, because the utility cost of a catastrophic loss exceeds the expected value calculation. It is why lottery tickets are bought — tiny probabilities of large wins can have high utility for people in poor financial situations even when the expected value is negative.

Later solutions

Utility theory was not the last word. Later thinkers pointed out that no casino actually has infinite wealth — in reality the prize is bounded by the casino's total assets. Behavioural economists noted that humans systematically underweight very small probabilities, which also resolves the paradox descriptively if not normatively. The paradox remains one of the richest problems in the philosophy of probability — a puzzle about what it means to be rational under uncertainty.