The St. Petersburg Game

Question

A fair coin is flipped until the first head appears. If the first head is on flip n, you are paid 2^n. What is the expected payout, and, more importantly, what would you actually pay to play once? Defend the gap.

Accepted Answer

The expectation diverges: mathbbE = sum_n=1^infty underbrace2^-n_P(textfirst head at n) cdot 2^n = sum_n=1^infty 1 = infty. Yet nobody sane pays more than 10–30 to play. A strong interview answer names several independent reasons rather than hand-waving "infinity is weird": - Counterparty cap. If the house can pay at most 2^30 (~a billion), the EV truncates to about 30. Your true expectation is against a credible payout distribution, not a mathematical one. - Concave utility / log wealth. Under log utility the certainty equivalent is modest (tens of dollars for reasonable wealth) because the enormous payouts arrive with vanishing probability. - One-shot variance. The median payout is 2–4; the game's value is carried by astronomically rare tails. For a single play, a fee above ~10 loses with overwhelming probability, and you can't iterate enough to reach the long run. Trader takeaway: EV is the start of pricing, not the end. Tail credibility, risk aversion, and repetition structure determine what an expectation is worth, the same triage you'd run on any too-good-looking trade.

The St. Petersburg Game

Problem

Your answer

Hints

More Expected Value to train on