St. Petersburg paradox
A bet with infinite expected value that almost no one will pay much to play.
What it means
The St. Petersburg paradox is a gamble — a coin is flipped until heads, paying 2^n ducats if the first head appears on flip n — whose expected monetary value is infinite, yet which people will pay only a few coins to enter. The paradox motivated the very idea of utility: Bernoulli argued that people maximize expected utility, not expected money, and that the logarithmic shape of utility makes the bet's worth finite. It thus marks the historical birth of diminishing marginal utility and of expected-utility theory itself. Modern treatments note that bounded utility, finite bankrolls, and probability weighting all further deflate its value. The puzzle endures as the clean illustration that expected value alone cannot describe how people value risk.
Examples
Asked what they would pay to enter a game with literally infinite average payout, most people offer only a handful of dollars.
Run the doubling game for real and the house is bankrupted by the first long run of tails; people price the bet at what could actually be paid, not at its average.
Bernoulli's answer is why a lost £100 stings a student far more than a millionaire: each extra pound adds less satisfaction, so even an unlimited average payout is worth only coins.
First described in Nicolas Bernoulli (1713); resolution by Daniel Bernoulli (1738).