The St. Petersburg Paradox

Five families of response, and where each one breaks.

1. Diminishing marginal utility (Daniel Bernoulli, 1738; Gabriel Cramer, 1728). Value money by a concave function. Cramer, in a letter to Nicolaus Bernoulli a decade before Daniel's paper, suggested the square root, giving a value of about 5.83 ducats; Daniel used the logarithm and got 4. Breaks against: Menger's super-St-Petersburg game. Any unbounded utility function can be defeated by growing the prizes fast enough.

2. Bounded utility (Cramer, 1728; Menger, 1934; Savage, 1954). Insist that utility has a ceiling. This genuinely works — it is the only response that makes the expectation finite for every lottery — and it is the orthodox position in decision theory, entrenched by Savage's axiomatisation. The cost: you must accept that there is some quantity of good beyond which nothing can be better. Critics find that hard to swallow as a claim about value rather than about psychology.

3. Finite bankrolls and finite time (Buffon, 1777; many since). No bank can pay unbounded prizes, and no game can run for unbounded time. Truncate and the value drops to tens of pounds. The objection: this dissolves the practical question but not the theoretical one. Decision theory is supposed to tell you what to do in principle, and "the premise is physically unrealisable" is a dodge if the premise is coherent.

4. Reject expected value as the criterion. Perhaps maximising expectation was never the right rule for lotteries with undefined or infinite means. Nicolaus Bernoulli himself leaned this way, suggesting that outcomes below some probability threshold should simply be ignored. The objection: every proposed threshold is arbitrary, and abandoning expectation costs you the theorems that make decision theory work.

5. Time averages and ergodicity (Ole Peters, 2011). The most recent serious entrant. Peters argues that the expected value averages over parallel possible worlds — an ensemble average — while an actual gambler experiences a single sequence through time. For a multiplicative wealth process these two averages are not equal, because the process is not ergodic. Compute the time-average growth rate instead and you get a finite, wealth-dependent fair price, with no utility function required.

Where Peters is contested. The resulting decision rule is mathematically the Kelly criterion, and it produces the same number as Bernoulli's logarithm. Critics — Jayanth Varma's 2013 rebuttal is the most direct — argue that this makes it a rederivation rather than a resolution: logarithmic utility, arrived at by a different route and with a different story attached. Peters' reply is that the story is the point, because it explains why the logarithm rather than positing it. Whether that counts as progress is a genuine open disagreement, not a settled matter.

Why the nature is "apparent contradiction". There is no contradiction in the mathematics: the sum diverges, full stop. The contradiction is between a principle (maximise expected value) and a judgement (don't pay much), and it dissolves once you name the hidden assumption — variously identified as unbounded utility, an infinitely rich counterparty, unlimited time, or ergodicity. What keeps the paradox alive is that different people name different assumptions, and no one naming has won.

9 September 1713. Nicolaus Bernoulli wrote to Pierre Rémond de Montmort with several problems about games of chance. One of them was the coin-doubling wager. Montmort included the correspondence in the second edition of his Essay d'analyse sur les jeux de hazard the same year. Neither man resolved it. Nicolaus' inclination was that outcomes of sufficiently small probability should simply be disregarded — a position that has never quite gone away and has never been made principled.

1728. Gabriel Cramer, writing to Nicolaus from London, proposed both of the answers that would dominate the next three centuries: value money by its square root, or cap its value entirely. "Mathematicians value money in proportion to its quantity," he wrote, "and men of good sense in proportion to the usage that they may make of it." Cramer's letter predates Daniel Bernoulli's publication by a decade and is often under-credited.

1738. Daniel Bernoulli — Nicolaus' cousin, then working at the Imperial Academy of Sciences in St Petersburg — published Specimen Theoriae Novae de Mensura Sortis in the Academy's Commentarii. He proposed that "the utility resulting from any small increase in wealth will be inversely proportionate to the quantity of goods previously possessed", which integrates to a logarithm, and showed that this makes the wager's value finite. The paradox takes its name from the journal, not from any events in the city. The paper was translated into English by Louise Sommer only in 1954, for Econometrica.

1777. Georges-Louis Leclerc, Comte de Buffon, did what nobody had thought to do: he ran the experiment. He had a child flip coins for 2,048 plays of the game and reported the empirical average — a little over 4 units per game, and no catastrophic payout. He also pressed the finite-bankroll objection.

1934. Karl Menger published "Das Unsicherheitsmoment in der Wertlehre", demonstrating the super-St-Petersburg construction. This is the single most consequential technical result on the problem: it converted "use a concave utility function" from a solution into a partial one, and made boundedness the issue.

1944–1954. Von Neumann and Morgenstern's Theory of Games and Economic Behavior and Savage's The Foundations of Statistics built the axiomatic framework of modern decision theory. Within it, bounded utility became the standard response, and the St Petersburg game became the standard argument for it.

1977. Paul Samuelson published "St. Petersburg Paradoxes: Defanged, Dissected, and Historically Described", the definitive survey. His conclusion was that the paradox has been resolved repeatedly because it is several problems wearing one name.

2011. Ole Peters published the time-resolution argument in Philosophical Transactions of the Royal Society A, launching the ergodicity-economics programme. The debate that followed is ongoing.

The St Petersburg paradox produced the concept of utility. That alone would secure its place: diminishing marginal utility is the load-bearing idea beneath insurance, progressive taxation, portfolio theory, consumer demand and most of welfare economics. It was invented, in 1738, to explain why nobody wants to buy a coin-flipping game.

Beyond that, the paradox is the standard worked example of three distinct lessons.

Averages can fail to exist, and the failure is not exotic. Distributions with infinite or undefined means are not mathematical curiosities. Wealth, city sizes, insurance claims, word frequencies, network degrees, pandemic sizes and the returns of venture capital funds all have heavy tails, and several are close to the regime where sample means do not converge usefully. Reporting the mean of such a quantity is not merely imprecise — it is reporting a number that describes nothing.

"Maximise expected value" is a rule with conditions attached. It is a theorem under specific axioms, not a self-evident maxim. When the expectation diverges, the theorem does not apply, and the intuition that says "pay whatever it costs" has no support.

The distinction between the average outcome and the typical outcome is practical, not philosophical. A strategy with excellent expected value and a high probability of ruin is a bad strategy, and this is the same observation that underlies the Kelly criterion, position sizing, and every insurance contract ever written.

Heavy tails in finance. Value-at-risk models, catastrophe insurance pricing and hedge-fund return distributions all live in the territory where the mean is either infinite or so poorly estimated as to be useless. The 2008 crisis produced a large literature on strategies whose expected value looked excellent and whose time-average behaviour was ruinous — precisely the St Petersburg distinction.

Kelly betting and position sizing. The Kelly criterion maximises the long-run growth rate rather than the expected return, and it is standard practice in professional gambling and parts of quantitative finance. It is Bernoulli's 1738 answer, operationalised, and Peters' formulation makes the connection explicit.

Existential risk and longtermist ethics. Arguments of the form "this intervention has a tiny probability of an astronomically large payoff, so its expected value dominates everything" — sometimes called Pascalian mugging — are St Petersburg arguments. The standard replies are the standard replies: bounded value, discounting negligible probabilities, refusing to use expectations for undefined-mean prospects. The debate in effective-altruism circles is a direct continuation of the eighteenth-century one.

Reinforcement learning. Agents maximising expected discounted reward in environments with heavy-tailed returns can be driven by a vanishingly rare outcome. Distributional RL, risk-sensitive objectives and CVaR-based criteria are engineering responses to exactly this failure mode.

Ergodicity economics. Peters' programme has produced a body of work applying the time-average framework to insurance, cooperation, leverage and inequality, and there is now experimental work on whether human risk preferences track time-average optimality. Whether it constitutes a new foundation or a repackaging of expected utility is genuinely contested — which is itself the clearest sign that this 300-year-old puzzle is not finished.

The divergent expectation

Let $X$ be the payout. The first head occurs on flip $k$ with probability $2^{-k}$, and pays $2^{k}$:

$$\mathbb{E}[X] = \sum_{k=1}^{\infty} 2^{-k}\cdot 2^{k} = \sum_{k=1}^{\infty} 1 = \infty$$

Every term is exactly 1. The series is not conditionally convergent, not rearrangement-sensitive, not subtle: it is $1 + 1 + 1 + \cdots$.

The distribution has infinite mean but a perfectly ordinary median of 2, and $P(X \geq 2^{m}) = 2^{-(m-1)}$ — a power-law tail with index 1, exactly the borderline case where the mean fails to exist.

Bernoulli's logarithmic utility

Take $u(w) = \ln w$ and a player with initial wealth $w$ paying a stake $c$. Expected utility after playing:

$$\mathbb{E}[u] = \sum_{k=1}^{\infty} 2^{-k},\ln!\left(w - c + 2^{k}\right)$$

This converges, because $\ln(2^k) = k\ln 2$ grows linearly while $2^{-k}$ decays geometrically. The fair stake $c^{*}$ is the value making $\mathbb{E}[u] = \ln w$.

In the simplest case ($w = 0$, valuing the prize alone):

$$\mathbb{E}[\ln X] = \sum_{k=1}^{\infty} 2^{-k}, k \ln 2 = \ln 2 \sum_{k=1}^{\infty} k,2^{-k} = 2\ln 2 = \ln 4$$

The certainty equivalent is exactly 4. Cramer's square-root utility gives

$$\mathbb{E}[\sqrt{X}] = \sum_{k=1}^{\infty} 2^{-k} 2^{k/2} = \sum_{k=1}^{\infty} 2^{-k/2} = \frac{1}{\sqrt{2}-1} \approx 2.414$$

with certainty equivalent $2.414^{2} \approx 5.83$.

Note that both answers are small, both are finite, and both land in the range people report being willing to pay. That empirical fit is why the idea took hold.

Menger's super-St-Petersburg game

Let $u$ be any utility function that is unbounded above. Then for each $k$ there exists a prize $x_{k}$ with

$$u(x_{k}) \geq 2^{k}$$

Construct a lottery paying $x_{k}$ with probability $2^{-k}$. Its expected utility is

$$\mathbb{E}[u] = \sum_{k=1}^{\infty} 2^{-k}u(x_{k}) \geq \sum_{k=1}^{\infty} 1 = \infty$$

and the paradox has returned in identical form.

For logarithmic utility, take $x_{k} = e^{2^{k}}$ — a prize that is doubly exponential in $k$. For any $u$ with $u(x) \to \infty$, some such sequence exists.

Menger's conclusion: utility must be bounded above. This is not a convenience; it is forced. Von Neumann–Morgenstern and Savage's axiomatisations both accommodate it, and boundedness is the orthodox position — Arrow argued in 1970 that the continuity axiom together with the existence of St Petersburg-type lotteries essentially requires it.

The price. Bounded utility implies there is a level of goodness that cannot be exceeded, no matter what. It also implies your preferences between very large outcomes become almost flat: a certainty of $10^{100}$ lives saved and a certainty of $10^{200}$ are nearly indifferent. Many find that a reductio of bounded utility rather than a resolution of the paradox. The literature has not settled this.

Finite bankrolls, precisely

Suppose the house can pay at most $2^{L}$, and pays $2^{L}$ if the head has not appeared within $L$ flips. Then

$$\mathbb{E}[X] = \sum_{k=1}^{L} 2^{-k}2^{k} + 2^{-L}2^{L} = L + 1$$

So the fair price is logarithmic in the bank's wealth $W$: $\mathbb{E}[X] \approx \log_{2}W + 1$.

This has a pleasing consequence. Under Bernoulli's log utility with wealth $w$, the fair price is also roughly logarithmic. The finite-bankroll argument and the log-utility argument produce the same functional form by completely independent routes — one from the physics of the counterparty, one from the psychology of the player. Samuelson's 1977 survey treats this convergence as the reason the paradox keeps being declared solved and keeps returning: several distinct problems share one symptom.

Feller's weak law

The distribution has no mean, but the sample average is not lawless. Feller proved that for $S_{n}$ the total of $n$ independent plays,

$$\frac{S_{n}}{n\log_{2}n} \xrightarrow{\ \mathbb{P}\ } 1$$

So the average payout after $n$ games is approximately $\log_{2}n$: about £20 after a million games, £40 after a trillion. The expectation diverges, but at a rate no gambler can exploit and no experiment can observe.

This gives the sharpest statement of the practical resolution: a price is fair for a game you will play $n$ times if it is near $\log_{2}n$. Nothing about the infinite expectation is relevant to any finite life.

Peters and the time average

Peters starts from the observation that wealth compounds multiplicatively, so the relevant question is not "what is the average of $w_{t+1}$ over possible outcomes?" but "what is the growth rate of $w_t$ along a single trajectory?"

For a player of wealth $w$ paying $c$ per round, the time-average exponential growth rate is

$$\bar{g}(c) = \sum_{k=1}^{\infty} 2^{-k},\log_{2}!\left(\frac{w - c + 2^{k}}{w}\right)$$

This converges. The maximum acceptable stake is the $c^{}$ solving $\bar{g}(c^{}) = 0$: pay more and your wealth shrinks in the long run, pay less and it grows.

Three features are worth stating plainly:

1. $c^{*}$ is finite, so the paradox does not arise.
2. $c^{*}$ depends on your wealth $w$ — a richer player rationally pays more, with no appeal to psychology or taste.
3. No utility function appears anywhere. The logarithm arrives from the multiplicative dynamics, not from an assumption about value.

The gap between $\bar{g}$ and $\mathbb{E}$ is the non-ergodicity of the process: the ensemble average over parallel worlds is dominated by a vanishing fraction of them, while the time average tracks the typical trajectory. Peters argues the same diagnosis explains risk aversion, insurance and leverage limits generally.

The rebuttal, stated fairly. Varma (2013) and others point out that maximising $\bar{g}$ is precisely the Kelly criterion, that Kelly is observationally equivalent to logarithmic utility, and that Peters therefore recovers Bernoulli's 1738 answer with a new justification rather than solving anything Bernoulli left open. Peters' response is that a derivation is worth more than a postulate: log utility was chosen in 1738 because it worked, whereas the time average explains why the logarithm is the right function. Readers can judge; the literature has not converged.