ParadoxesXII
The St. Petersburg paradox
A coin-flipping game has an infinite expected payout. Yet almost no one would pay more than a few dollars to play it. Who is irrational: us, or the maths?
- Reading time
- 4 min
- Difficulty
- Intermediate
- Branch
- Decision
- First posed
- Nicolaus Bernoulli, 1713; solved in one way by Daniel Bernoulli, 1738, published in St. Petersburg
The puzzle
A casino offers you a game. A fair coin is flipped until it lands heads. If heads comes up on the first flip, you win 2 dollars. If it first comes up on the second flip, you win 4 dollars. On the third, 8 dollars. Each time tails comes up, the pot doubles. How much should you pay to play?
The standard way to value a gamble is its expected value: add up each possible prize multiplied by its probability. There is a 1/2 chance of 2 dollars, contributing 1 dollar. A 1/4 chance of 4 dollars, contributing 1 dollar. A 1/8 chance of 8 dollars, contributing 1 dollar. Every possible outcome contributes 1 dollar, and there are infinitely many outcomes. The expected value is infinite.
So, by that reasoning, you should be willing to pay any finite amount to play: your house, your savings, everything. Yet when people are asked, most would pay only a few dollars. Half the time, after all, you win just 2 dollars.
The puzzle was posed by Nicolaus Bernoulli in 1713. His cousin Daniel Bernoulli proposed a famous solution in 1738, in a paper published by the Imperial Academy of Sciences in St. Petersburg, which gave the puzzle its name.
Why it matters
The paradox launched the concept of utility and the idea of diminishing marginal value, which became foundations of modern economics. It shows that maximising expected money is not the same as maximising what you care about.
It also exposes deep problems in decision theory when outcomes can be unboundedly large, problems that resurface in Pascal's mugging and in reasoning about long-term and existential risks.
Key terms
- Expected valueThe average result you would get if you repeated a gamble many times, found by multiplying each payoff by its probability and adding.
- UtilityA measure of how good an outcome is for someone, used in ethics and decision theory. It need not be the same as money or pleasure.
- Diminishing marginal utilityThe principle that each additional unit of something, such as money, adds less value than the one before.
- Risk aversionPreferring a certain outcome to a gamble with the same or even higher expected value.
- Bounded utilityThe view that there is an upper limit to how good any outcome can be for a person, so that no promised reward can be large enough to outweigh every tiny probability.
Proposed solutions
Diminishing marginal utility
Money is worth less to you the more you have. Measured in utility rather than dollars, the game's expected value is modest.
The casino cannot pay
No real casino could cover the large prizes. Once you account for its finite bankroll, the expected value is small.
Ignore tiny probabilities
Outcomes with very small probability should be disregarded. Once the long tail is ignored, the game is worth little.
Expected value is the wrong measure
Rational choice should consider more than the average. Risk, variance and the probability of losing matter too.
Each solution, tested
Diminishing marginal utility
- How it resolves the paradox
- Daniel Bernoulli argued that a thousand dollars means far more to a poor person than to a rich one. If utility grows only with the logarithm of wealth, each doubling of the prize adds a fixed amount of utility, while its probability halves. The expected utility of the game is then finite and small, matching what people would actually pay. This idea, of diminishing marginal utility, became a pillar of economics.
- Why it doesn’t fully work
- Karl Menger showed in 1934 that the paradox can be revived: just make the prizes grow fast enough that each one adds the same amount of utility, not money. Any unbounded utility function admits a 'super St. Petersburg' game with infinite expected utility.
- What defenders say
- This pushes some theorists toward bounded utility, where there is a maximum to how much anything can be worth to you.
The casino cannot pay
- How it resolves the paradox
- If the casino has, say, a billion dollars, all prizes above that are capped. The game then has only around thirty possible distinct payouts before the cap is reached, and its expected value is around thirty dollars. The infinite value only arises in an impossible idealisation.
- Why it doesn’t fully work
- This solves the practical problem but not the theoretical one. The paradox asks what a rational agent should pay for the idealised game, and the answer 'infinite' still seems wrong.
Ignore tiny probabilities
- How it resolves the paradox
- In practice, people treat sufficiently unlikely events as impossible. The Comte de Buffon suggested treating any probability below about 1 in 10,000 as zero. Applied here, all the huge prizes, which are extremely unlikely, are dropped, and the game is worth only a few dollars.
- Why it doesn’t fully work
- Any threshold is arbitrary, and discounting low-probability outcomes can lead to bad decisions when many such outcomes add up, or when a rare outcome is catastrophic.
Expected value is the wrong measure
- How it resolves the paradox
- Expected value is an average over infinitely many repeated plays, but you play once. Lara Buchak argues that rational people can legitimately weigh the worst outcomes more heavily than the best. The St. Petersburg game is a bad bet because its value is concentrated in outcomes you will almost certainly never see.
- Why it doesn’t fully work
- Departing from expected utility opens the door to inconsistent preferences, which a clever bookmaker might exploit. Defenders of expected utility argue that its consistency is precisely its virtue.
Who argued what
- Diminishing marginal utility Daniel Bernoulli, Gabriel Cramer
- The casino cannot pay Georges-Louis Leclerc, Comte de Buffon, William Feller
- Ignore tiny probabilities Georges-Louis Leclerc, Comte de Buffon, Bradley Monton
- Expected value is the wrong measure Lara Buchak, Maurice Allais
Variations
The Pasadena game
Harris Nover and Alan Hájek devised a variant in which payoffs alternate between positive and negative, chosen so that the expected value depends on the order in which you add up the outcomes. It can be made to equal anything, or nothing. If a game has no well-defined expected value at all, how should you value it?
The simulation
If you simulate the game a million times, the average payout tends to grow very slowly, roughly with the logarithm of the number of games played. In any realistic number of plays, the average stays modest. The infinite expectation lives in the possibility of enormously rare events.
Go deeper
How the cousins disagreed
Nicolaus Bernoulli posed the problem in a letter to Pierre Rémond de Montmort. Gabriel Cramer suggested in 1728 that money's value increases less than proportionally, anticipating Daniel's solution. Nicolaus himself did not accept the utility solution, and suggested instead that very small probabilities should be neglected. The family debate set out most of the responses still discussed today.
Related ideas
Further reading and sources
- EncyclopediaStanford Encyclopedia of Philosophy: The St. Petersburg Paradox
- WikipediaSt. Petersburg paradox
- BookRisk and Rationality, Lara Buchak
- BookAgainst the Gods, Peter L. Bernstein
Open questions
- How much would you actually pay to play, and why?
- Is a gamble with an infinite average worth an infinite price?
- Should you ever bet everything on an outcome you will almost certainly never see?