St. Petersburg Paradox

The game

The rules are simple. A fair coin is flipped repeatedly until the first tail. Let n be the flip number on which the first tail occurs. Your prize is 2ⁿ dollars. So:

• Tail on flip 1 (probability 1/2) → $2.
• Tail first on flip 2 (probability 1/4) → $4.
• Tail first on flip 3 (probability 1/8) → $8.
• Tail first on flip n (probability 1/2ⁿ) → $2ⁿ.

The expected payoff is

E[payoff] = Σ_{n=1}^∞  (1/2)^n · 2^n  =  Σ_{n=1}^∞  1  =  ∞

Every term in the sum is 1: probability shrinks by half each step, but the payoff doubles. Infinitely many terms of size 1 sum to infinity. A risk-neutral expected-value maximizer should pay any finite price to play. Yet experimental subjects bid $5–$25 typically; auctions with real prizes rarely close above $20 even when the experimenter caps the game length so the mean is well-defined.

Bernoulli's resolution with log utility

Daniel Bernoulli's 1738 insight: subjective value rises with wealth, but with diminishing returns. The second million dollars matters less than the first. He proposed u(W) = log(W).

Suppose your wealth before playing is W. Expected log-wealth after paying entry fee c and playing:

E[log(W − c + 2^n)] = Σ_{n=1}^∞  (1/2)^n · log(W − c + 2^n)

You should play if this exceeds log(W). The certainty-equivalent fair price c* is the c that makes the inequality bind. For W = $1,000, c* ≈ $6. For W = $1,000,000, c* ≈ $20. Bernoulli's calibration is remarkably close to real-world bids — though it underweights how much variance from his theoretical W=∞ player would shift the bid.

To see why log utility works: the marginal utility of an extra dollar is 1/W, so each doubling of payoff yields the same incremental utility (log doubles by log(2) ≈ 0.69, a constant). The exponential growth of payoffs no longer compensates for the geometric decline in probability, and the expected utility sum converges.

Bernoulli, von Neumann, Markowitz: same idea, different framings

Approach	Mathematical object	Resolution mechanism	Date
Risk-neutral EV	Σ pn · xn	Fails — sum diverges	—
Bernoulli (1738)	Σ pn · log(W + xn)	Concave u; convergent	1738
Cramer (cited by Bernoulli)	Σ pn · √xn	Concave u with sqrt; convergent	~1728
Menger bounded utility	min(x, C) replacement	Hard cap removes super-Petersburg variants	1934
vNM expected utility	Σ pn · u(xn) for general concave u	Axiomatized version of Bernoulli	1944
Kelly criterion	Maximize E[log(W)] long-run growth	Same log structure for repeated betting	1956
Prospect theory	Σ w(pn) · v(xn) — value coded as gains	Probability weighting + value curvature	1979
Bounded vNM utility	u(x) ≤ C ∀ x	Resolves arbitrary super-Petersburg	Modern consensus

From Basel to St. Petersburg to Princeton

The puzzle originated in a 1713 letter from Nicolas Bernoulli (mathematician, Basel) to Pierre Rémond de Montmort, a French probabilist. Nicolas's framing concerned moral hazard in dice games — but the underlying paradox was the same. Daniel Bernoulli, Nicolas's cousin, took up a research post at the Imperial Academy of Sciences in St. Petersburg in 1725 and published the log-utility resolution there in 1738. The Latin journal was the Commentarii Academiae Scientiarum Imperialis Petropolitanae — hence the name. The paper, "Specimen Theoriae Novae de Mensura Sortis" ("Exposition of a New Theory on the Measurement of Risk") was 21 pages and laid out essentially the modern expected-utility framework, including the formal definition of risk aversion via concave u.

The paper was ignored for 200 years. In 1934 the Austrian mathematician Karl Menger (son of the economist Carl Menger) re-introduced it to the economics literature with the influential note "Das Unsicherheitsmoment in der Wertlehre" — a paper that catalogued how each proposed u(·) function could be broken by a sufficiently rapidly growing payoff and concluded that any complete resolution required bounded utility. Von Neumann and Morgenstern's 1944 axiomatization (the EU theorem) implicitly handled the convergence question through their finite-outcome assumption.

Modern asset pricing — Lucas 1978, Merton 1969 continuous-time consumption — uses the log-utility special case directly: CRRA(γ=1) is log, and most asset-pricing intuitions are taught via log utility because the closed-form solutions are clean.

Variants and extensions

• Super-St. Petersburg. Replace the payoff 2ⁿ with 2²ⁿ, so even log utility gives an infinite expected utility. Only utility bounded above resolves arbitrary growth rates (Menger 1934).
• Truncated game. Cap the game at N flips. The EV becomes N + finite tail, easy to compute. At N = 30 flips, EV ≈ $31. Real-world risk aversion to the 30-flip game is much milder than to the unbounded game.
• Pasadena game (Nover-Hájek 2004). Alternating positive and negative payoffs whose expectation is undefined (not even ±∞). Shows the paradox isn't really about infinity but about ill-defined limits.
• Kelly betting. John Kelly's 1956 paper showed that maximizing E[log(wealth)] is the unique strategy that maximizes long-run growth rate in repeated independent gambles — the dynamic version of Bernoulli's static log utility.
• Cumulative prospect theory restatement. Probability weighting w(p) underweights small probabilities of huge payoffs, dampening the infinite mean computationally — closer to actual bids than expected utility alone.
• Variance-sensitive reformulation. Allais argued the variance of the lottery matters too; the St. Petersburg game has infinite variance even with finite Bernoulli-utility expectation, plausibly driving down willingness-to-pay.

Where the paradox shows up in 2025

• Venture capital returns. Power-law distributed outcomes — most startups fail, a tiny fraction return 10,000×. Standard NPV underestimates portfolio value when the tail is heavy; log-utility valuation models are now standard at YC, Sequoia, etc.
• Catastrophe-linked securities. Cat bonds, pandemic bonds, and parametric insurance all have tail payoff structures formally similar to St. Petersburg, with rare enormous payouts. Pricing models use concave-utility certainty equivalents.
• Lottery sales. US Powerball jackpots regularly cross $1B with 1-in-300M odds — EV per dollar is negative, but the tail-weighted preference for huge unlikely payoffs (the inverse Bernoulli phenomenon) drives $113B/year sales.
• Pascal's wager. Pascal's 1670 argument for belief in God — infinite payoff with positive probability — is the theological version of St. Petersburg. Modern responses use bounded utility or many-gods objections.
• Climate catastrophe pricing. Weitzman's 2009 "dismal theorem" showed that fat-tailed temperature distributions make standard cost-benefit analysis explode, much like St. Petersburg. Bounded-utility versions of integrated assessment models are now standard.
• AI x-risk economics. Bostrom-style arguments about tiny probabilities of existential payoffs (positive or negative) inherit the St. Petersburg structure. Bounded utility is a standard counter-objection.

Common pitfalls in interpreting the paradox

• Treating finite-time variants as the same paradox. Capping at 30 or 50 flips gives a finite mean ($30 or $50) — the paradox is specifically about the infinite-horizon game.
• Forgetting that variance matters too. Even after taking logs, the standard deviation of payoff (in log-dollars) remains large. A risk-averse agent considers more than just E[u].
• Assuming Bernoulli's price calibration is universal. Bernoulli's specific c* values depend on starting wealth W and the log functional form. Replacing log with CRRA(γ=2) gives different and lower fair prices.
• Confusing this with the gambler's ruin problem. Gambler's ruin concerns repeated bets with a finite bankroll; St. Petersburg is a one-shot game. Different mathematical structure entirely.
• Skipping the super-Petersburg counterexample. Any specific concave u (log, sqrt, etc.) is broken by a fast-enough growing payoff. Only bounded u resolves every variant — Menger's lesson.