Thank you for visiting this site. This article covers “The St. Petersburg Paradox.”
If the expected value of a game is mathematically infinite, how much would you pay to play? $100? $10,000? $1,000,000? In theory, no entry fee is too high — “in the long run you can’t lose.” Yet in practice almost everyone would pay only a few dollars. This gap between intuition and theory is the heart of a paradox that has been debated for over 300 years.
The Rules
The rules are simple.
Flip a fair coin until heads appears. The prize is determined by which flip produced the first heads.
- Heads on flip 1 → prize: $2
- Heads on flip 2 → prize: $4
- Heads on flip 3 → prize: $8
- Heads on flip 4 → prize: $16
- Heads on flip n → prize: $2ⁿ
The later the first heads, the larger the prize doubles.
Computing the Expected Value
Expected value is the average payoff over many plays.
- First flip is heads (probability 1/2), prize $2 → contribution: 1/2 × 2 = $1
- Second flip is the first heads (probability 1/4), prize $4 → contribution: 1/4 × 4 = $1
- Third flip is the first heads (probability 1/8), prize $8 → contribution: 1/8 × 8 = $1
- …
Every stage contributes exactly $1 to the expected value. Since this continues infinitely, Expected value = 1 + 1 + 1 + 1 + … = infinity.
An infinite expected value means that, in theory, no entry fee — $1 million, $1 billion — is too much to pay. You should always play, because in the long run you come out ahead. Yet essentially no one would pay even $100 for a single play.
Why Intuition Says No
Thinking it through, there are several reasons.
First, the probability of a large prize is astronomically small. Getting 20 consecutive tails has probability roughly 1 in a million. To win more than $1 million, you need million-to-one luck.
In most cases — 75% of the time — the prize is $4 or less. If you pay $100 to play, you lose money on three plays out of four.
The expected value is infinite because an astronomically small probability of an astronomically large prize keeps contributing $1 per term forever. But in any realistic number of plays during a human lifetime, the outcome is almost certain to be far below infinity. The limit “works” only over a number of trials that will never be reached.
Bernoulli’s Solution: Utility Theory
The first resolution was proposed by Swiss mathematician Daniel Bernoulli in 1738 (presented at the St. Petersburg Academy of Sciences — hence the name).
Bernoulli argued that people do not evaluate outcomes in terms of money alone, but in terms of the utility (satisfaction) that money provides.
For a person who already has $1 million, receiving another $1 million is not twice as satisfying as it was to receive the first million. The utility of money increases at a diminishing rate (roughly logarithmically) as amounts grow larger.
Replacing dollar amounts with their utilities in the expected value calculation produces a finite result. This caps the reasonable entry fee at some finite amount — the paradox dissolves.
The Debate Continues
Bernoulli’s utility theory is widely accepted as one resolution, but it is not final. If you modify the game to pay out prizes that grow even faster than a person’s utility diminishes, the paradox returns.
Modern economics and decision theory analyze the St. Petersburg Paradox through multiple lenses: risk aversion, finite lifespans, diminishing marginal utility, and more.
The most lasting lesson may be that expected value alone is an insufficient guide to rational decision-making.
Summary
This article covered “The St. Petersburg Paradox.”
Proposed three centuries ago, this paradox reveals the gap between mathematical probability and real human decision-making. The fact that people refuse to pay large amounts for a game with infinite expected value may actually reflect something right about human judgment — not a failure of rationality.
To return to the full list of paradoxes, follow the link below.
Thank you for reading. We hope to see you in the next article.
