Cobra Effect · Statistics and probability
The St Petersburg paradox
A huge prize is worth less to you than its average suggests.
6 cards, read aloud in 2:52, with a test and sources.
A coin game. How much would you pay to play?
I toss a coin. If it lands heads, I pay you $2 and the game ends. If tails, the pot doubles and I toss again. Heads on the second toss pays $4, on the third $8, on the fourth $16. The game only ends on heads. Name your price.
Most people say a few dollars. The maths says infinity.
Half the time you win $2. A quarter of the time, $4. An eighth, $8. Each outcome is worth exactly one dollar of expectation, and there is no end to them. Add them up and the fair price is infinite. Yet nobody sane would pay $50.
Nicolas Bernoulli posed it in 1713. His cousin Daniel answered in 1738.
Nicolas put the puzzle in a letter to the mathematician Pierre Rémond de Montmort. Daniel Bernoulli published the answer in the journal of the St Petersburg academy, which gave the paradox its name. His idea was that a dollar is not worth a dollar. It is worth what it does for you, and that shrinks the more you already have.
That is diminishing utility, and it is the heart of the paradox.
The second million means less than the first. The thousandth dollar less than the tenth. So a one in a million chance of a vast fortune is not worth a millionth of the fortune. Value the winnings by what they mean rather than what they count, and the infinite sum collapses to a few dollars. Gabriel Cramer had reached a similar answer in 1728, and Daniel said so.
There is a plainer answer too. The bank runs out.
Nobody can pay an infinite prize. If the richest bank on earth backs the game, the pot is capped at some number of doublings, and the fair price is a few dozen dollars. Both answers are true. The infinite value was never real. It came from pretending that money has no limit and no meaning.
What it teaches about any big number.
When a plan is justified by a huge payoff at a tiny chance, ask two questions. What is that payoff actually worth to me, not on paper but in my life? And can it actually be paid? It is why insurance is worth buying, and why nobody should bet the house on a long shot. Most expected values are honest. The ones that look infinite are telling you the sum has gone wrong.
Sources
- St. Petersburg paradox, Wikipedia. The game, the infinite sum, the 1713 letter, and Daniel Bernoulli’s 1738 paper. Search Exposition of a New Theory on the Measurement of Risk for the English translation.
- Against the Gods: The Remarkable Story of Risk, Peter L. Bernstein, 1996. The history of how people learned to think about risk, with a whole chapter on the Bernoullis and the coin game. Readable and full of characters.
- Expected utility hypothesis, Wikipedia. The theory that grew out of Bernoulli’s answer. How economists value uncertain outcomes, and the cases where real people refuse to behave as the theory says.
Nearby ideas
- Loss aversion. Losing something hurts about twice as much as gaining it pleases.
- The gambler’s fallacy. Chance has no memory, so nothing is ever due.
- Satisficing. Good enough, chosen quickly, often beats the endless hunt for best.
- Benford’s law. In real data, numbers starting with 1 far outnumber those starting with 9.
- The hot hand. Streaks are mostly chance, though not always entirely.
- The law of large numbers. Averages settle down over many tries, but a run never gets corrected.
- Mean and median. Why the average can describe nobody at all.
- The Literary Digest poll. A small sample chosen well beats a huge one of the wrong people.