So what is the game actually worth?

There is no single number — it depends on your utility and bankroll. Under log utility with a realistic bankroll, the fair price is small (often single digits to low tens), which matches what people will actually pay.

Is the St. Petersburg paradox still relevant to trading?

Very. It is the cleanest illustration of why traders optimise growth and survival rather than raw expected value — the same reasoning behind position sizing and the Kelly criterion.

The St. Petersburg Paradox

NeetQuant Team · June 2026 · 5 min read

Here is a game. A fair coin is flipped until it first lands heads. If the first head appears on flip $n$ , you are paid $2^n$ dollars. How much should you pay to play?

The infinite expected value

The probability the first head lands on flip $n$ is $1/2^n$ , and the payoff is $2^n$ , so every term of the expected value contributes the same amount:

$\mathbb{E}[\text{payoff}] = \sum_{n=1}^{\infty} \frac{1}{2^n}\cdot 2^n = \sum_{n=1}^{\infty} 1 = \infty.$

First head on flip	Probability	Payoff	Contribution to EV
1	1/2	2	1
2	1/4	4	1
3	1/8	8	1
n	1/2ⁿ	2ⁿ	1

Every row contributes 1, and there are infinitely many rows — hence the infinite sum. By a naive expected-value rule you should happily pay any finite amount to play. Yet almost nobody would pay even 20 dollars. That clash — infinite EV, tiny willingness to pay — is the St. Petersburg paradox, posed in the 1700s and still a favourite interview prompt.

Why the naive answer is wrong

Two ideas resolve it.

Diminishing utility. A dollar matters less the more you already have. If your satisfaction grows like $\log W$ rather than $W$ itself, the expected utility of the game is finite and small — Daniel Bernoulli's original 1738 resolution. Under log utility the fair price is modest, matching intuition.

Finite bankrolls and finite play. The infinite EV is driven entirely by astronomically rare, astronomically large payoffs — a first head on flip 40 pays over a trillion dollars. No casino can honour that, and you will never play enough times to realise the tail. Cap the payoff at any real bankroll and the sum collapses to a small number.

The quant connection

This is not a curiosity — it is the same lesson behind the Kelly criterion. Kelly sizes bets to maximise expected $\log W$ , exactly the utility that tames St. Petersburg.

The takeaway. Expected value is the wrong objective when payoffs are wildly skewed or when ruin is possible. Traders optimise for growth rate and survival, not the raw mean — which is why "what's the EV?" is so often followed by "and would you actually take this bet?"

Practise the reasoning

Get fluent with expected value and probability so you can compute the EV instantly — then be ready to explain why the EV is not the price. Firms like SIG and Jane Street probe exactly this judgement.

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Frequently asked questions

So what is the game actually worth?: There is no single number — it depends on your utility and bankroll. Under log utility with a realistic bankroll, the fair price is small (often single digits to low tens), which matches what people will actually pay.
Is the St. Petersburg paradox still relevant to trading?: Very. It is the cleanest illustration of why traders optimise growth and survival rather than raw expected value — the same reasoning behind position sizing and the Kelly criterion.