What is the St. Petersburg paradox?

A fair coin is flipped until it first lands heads. The player wins 2 to the power n dollars, where n is the number of flips. Heads on the first flip pays $2, on the second $4, on the third $8, and so on. The probabilities of these outcomes are one half, one quarter, one eighth, and so on. Each possible outcome therefore adds 1 to the expected value, and the sum grows without bound, as the [plato.stanford.edu](https://plato.stanford.edu/entries/paradox-stpetersburg/) entry and [en.wikipedia.org](https://en.wikipedia.org/wiki/St._Petersburg_paradox) both show. [1] [2]

Yet the likely result is modest. The probability is one half that the player wins no more than $2, and three quarters that the player wins no more than $4. Paying a very large fee to play seems absurd. Sources describe this as the discrepancy between the infinite expected value and what people will actually pay. Ian Hacking is quoted as saying few would pay even $25, and one popular explainer says people rarely offer more than $10. [1] [2] [3]

In a strict logical sense no contradiction is derived, so the Stanford Encyclopedia of Philosophy notes it is not a formal paradox. It is instead a apparent counterexample to maximizing expected value. The same entry builds a stronger version from three incompatible claims: the rational price is higher than every finite amount, the buyer knows the actual prize will be finite, and it is irrational to knowingly pay more than one will receive. The prizes are never infinite; only the expectation is unbounded. [1]

Daniel Bernoulli presented the problem to the St. Petersburg Academy in 1738. He argued that money should be judged by its usefulness, not its face amount, so large prizes add less value than their size suggests. He proposed that utility equals the logarithm of wealth, which builds in diminishing marginal utility. He also wrote that a reasonable person would gladly sell the chance for twenty ducats. Oxford Reference says the paradox led to his introduction of what later became called utility. [1] [2] [4]

Modern decision theorists agree this solution is too narrow. As the Stanford entry explains, following Menger in 1934, the prizes can be raised until they compensate for diminishing marginal utility of money. The game is then stated in units of utility, and its expected utility is again not finite. The prizes need not even be money. [1]