St. Petersburg Paradox, explained.
The St. Petersburg paradox concerns a lottery with infinite expected monetary payoff in its ideal unbounded form, despite finite willingness to pay to enter.
Why it happens
As the prize doubles, its probability halves, leaving a constant contribution from each possible stopping time. Infinitely many such contributions make the unbounded expectation diverge; a cap changes the game and makes the expectation finite.
The ideal unbounded St. Petersburg lottery has infinite expected monetary payoff, although willingness to pay is generally finite. A cap makes the expectation finite.
Read the result
Compare exact capped expectation with the mean and median of 200 samples. Raising the cap changes rare prizes much more than typical results. Resample to see why one batch can give a misleading sense of the average.
A worked example
An eight-flip cap
First heads pays 2, 4, 8 and so on, with a maximum of 256.
First-head outcomes on flips one through seven contribute seven units to expectation; the combined capped tail contributes two.
The exact mean is nine, while a small sample can miss the rare large payouts and have a different mean.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
At the first heads on flip k the payout is 2^k. If no heads occurs by cap n, the payout is 2^n. Each earlier first-head event contributes one to the mean; the capped tail contributes two, so the exact expected payout is n+1.
Where this idea is useful
A practical use
A headline average can give a poor picture of a typical outcome when rare extremes dominate it.
A common misconception
“Infinite expected value guarantees an enormous payment.”
Expectation weights possible outcomes; it is not a guaranteed or typical payout. A real finite cap also removes the infinite expectation.
What this explanation leaves out
- This finite capped version is not an infinite lottery. Sampled means vary and do not determine a fair price for every person.
Why does the final tail contribute two?
The probability of no heads in the first seven flips is 1/128, and all such histories pay 256 under the stipulated cap rule.
Does an average hide outcomes that are too rare for your experience or resources to absorb?
Further reading
Explore the original research or the teaching reference behind this experiment.