The St. Petersburg Paradox: When Infinity Meets Common Sense

In 1713, the Swiss mathematician Nicholas Bernoulli proposed a puzzle to his correspondent Pierre Rémond de Montmort. The puzzle has a simple setup. Imagine a casino offering this game:

Flip a fair coin repeatedly until it lands tails. If tails comes up on flip number $n$, the casino pays you $2^n$ dollars.

So if you get tails on the first flip, you win \2$. Heads-tails (tails on flip 2), you win$$4$. Heads-heads-tails,$$8$. Two heads then tails on flip 4,$$16$. Three heads then tails,$$32$. And so on.

How much would you pay to play this game?

Most people, asked this question, say something like “ten or twenty dollars.” Maybe a hundred if they’re being adventurous. Almost no one offers more than that.

But the mathematics says the expected value of the game is infinite. By the standard rule that you should be willing to pay any amount less than the expected payoff, you should be willing to pay \1{,}000$,$$1{,}000{,}000$, or any other amount to play. The mathematical analysis and human intuition disagree completely.

This is the St. Petersburg paradox, named after the city where Daniel Bernoulli (Nicholas’s cousin) published the first rigorous analysis in 1738. The paradox is one of the foundational problems in probability and decision theory. It led to the development of utility theory, which is now the standard mathematical framework for analyzing how rational people make decisions.

This article is about what the paradox says, why it’s genuinely puzzling, and how mathematicians and economists have tried to resolve it.

The expected value calculation

Let’s verify the infinite expected value claim. The probability of getting tails on flip $n$ (heads on the first $n-1$ flips, tails on flip $n$) is $(1/2)^n$. The payoff is $2^n$. So the contribution of outcome $n$ to the expected payoff is

$\frac{1}{2^n} \cdot 2^n = 1.$

Every term contributes exactly $1$ to the expected value. Summing over all $n \geq 1$:

$E[\text{payoff}] = 1 + 1 + 1 + 1 + \cdots = \infty.$

The series diverges. Each term, individually, is small ($1/2$ chance of \2$,$1/4$chance of$$4$, etc.) but there are infinitely many of them, each contributing the same amount.

By the expected utility hypothesis of classical economics — that rational agents should choose the option with the highest expected value — you should pay any finite amount to play this game. A million dollars, a billion, a googol. The math says go.

The math is correct. The recommendation is absurd. Why?

What’s actually likely to happen

The reason “go” feels absurd is that you’d almost certainly lose money. Let’s compute the distribution of possible payoffs:

• $50\%$ probability: tails on flip 1, win \2$.
• $25\%$ probability: tails on flip 2, win \4$.
• $12.5\%$ probability: tails on flip 3, win \8$.
• $6.25\%$ probability: tails on flip 4, win \16$.
• $3.125\%$ probability: tails on flip 5, win \32$.
• …

The median outcome is winning \2$— half the time, you only get$$2$. To win$$1{,}024$, you need 10 consecutive heads — about$0.1%$probability. To win$$1{,}000{,}000$, you need 20 consecutive heads — about$0.0001%$.

If you paid \1{,}000$to play once, you'd win less than that with probability$99.9%$. Almost certainly, you’d lose. Why does the expected value say you should pay so much?

The answer is that the expected value is dominated by extremely rare, extremely large payoffs. The contribution of “win \1{,}000{,}000{,}000$if you flip 30 heads" is$$1{,}000{,}000{,}000 / 2^{30} \approx $0.93$ — a real contribution to the expected value. But that scenario happens almost never. The mathematics treats all these rare-big payoffs seriously; human intuition discounts them heavily.

Daniel Bernoulli’s resolution

Daniel Bernoulli’s 1738 paper proposed a now-classical resolution: expected value is the wrong thing to maximize. You should maximize expected utility, where utility is a concave function of wealth.

The intuition: an extra dollar matters more to someone with \100$than to someone with$$1{,}000{,}000$. So the *utility* of money grows slower than money itself. Bernoulli proposed$u(W) = \log W$ — logarithmic utility — as a natural way to model this.

With log utility, the expected utility of the St. Petersburg game becomes:

$E[\log W] = \sum_{n=1}^{\infty} \frac{1}{2^n} \log(W_0 + 2^n - \text{fee}),$

where $W_0$ is your current wealth and the fee is what you paid to play.

This sum converges for finite $W_0$ and finite fee. With reasonable parameter values, Bernoulli’s calculation gives a fair fee of around \4$–$$25$ — much closer to what people actually offer. The contradiction between math and intuition is resolved.

This was one of the most important ideas in the history of economics. It introduced utility as a distinct concept from money, recognized that diminishing marginal utility matters, and made expected utility (rather than expected value) the right rationality criterion. Almost all of modern decision theory, microeconomics, and game theory is built on these foundations.

The visual story

Here’s why log utility tames the divergence:

The dashed line is “utility = money” — what you’d assume if every dollar mattered equally. The red curve is $u(W) = \log W$ — utility growing but at decreasing rate. With log utility, a payoff of \1{,}000{,}000{,}000$has roughly$9$units of utility; a payoff of$$1$has$0$. The 30 zeros of dollars don’t translate to 30 zeros of utility.

This concavity is what tames the St. Petersburg divergence. The contributions $\log(2^n)/2^n = n/2^n$ converge — the series $\sum n/2^n = 2$ has a finite value. With log utility, the St. Petersburg game has finite expected utility.

Other resolutions

Bernoulli’s resolution is the classical answer, but it’s not the only one. Several other resolutions have been proposed:

Bounded utility. Even simpler: utility functions in reality are bounded. There’s some maximum utility you could get from any payoff (you can’t enjoy \10^{100}$infinitely more than$$10^{50}$ — you’d be dead before you spent it). Any bounded utility makes the expected utility finite.

Risk aversion. Modern decision theory captures the same idea via the certainty equivalent: the guaranteed amount you’d accept instead of a gamble. For most people, the certainty equivalent of a coin-flip for \200$is less than$$100$ — risk aversion in action. Logarithmic utility is a specific functional form of risk aversion.

Finite resources of the casino. The casino cannot actually pay $2^{100}$ dollars. With a payoff cap of, say, \10^9$, the expected value is bounded by$\sum_{n=1}^{30} 1 = 30$. So the right fee is maybe$$20$, depending on the cap. This is the practical resolution: real games have finite stakes.

Heavy-tail aversion. Modern behavioral economics observes that humans are particularly averse to outcomes with heavy tails — distributions where extreme outcomes have non-negligible probability. The St. Petersburg game has a particularly heavy tail, and people’s reluctance reflects this specific aversion.

Each resolution captures a piece of why the St. Petersburg game feels different from what the expected value says. None is fully satisfactory by itself, but together they explain the gap.

The broader implications

The St. Petersburg paradox introduced several ideas that shaped centuries of subsequent economics and decision theory.

Expected utility theory (von Neumann and Morgenstern, 1944). Made expected utility the rigorous foundation of rational decision-making. Their axiomatization showed that any rational agent with consistent preferences must maximize expected utility for some utility function.

Risk aversion as a measurable property. Pratt (1964) and Arrow (1965) developed formal measures of risk aversion that economists still use. These measures distinguish people who avoid risk modestly from those who avoid it strongly.

Prospect theory (Kahneman and Tversky, 1979). A descriptive theory of how humans actually make decisions, accounting for cognitive biases the standard theory misses. Kahneman won the 2002 Nobel Prize in Economics for this work. Prospect theory predicts behavior in St. Petersburg-like situations more accurately than expected utility.

Heavy-tailed distributions as a real concern. Financial risk management, insurance pricing, and disaster planning all grapple with heavy-tailed payoffs. The St. Petersburg structure — small probability of enormous payoff — appears in real situations like venture capital investing, insurance against catastrophic losses, and lottery purchases.

The Kelly criterion. In repeated gambling situations, the optimal betting strategy maximizes log expected wealth — exactly Bernoulli’s resolution. Used by professional gamblers and quantitative traders.

The St. Petersburg paradox is, in this sense, a foundational paradox of modern economics. The 300-year-old puzzle continues to inform how economists, gamblers, investors, and decision theorists think about risk.

Generalizations

Variants of the original paradox produce interesting questions:

The triple St. Petersburg game: payoff is $3^n$ if tails first appears on flip $n$. Expected payoff: $\sum (1/2^n) \cdot 3^n = \sum (3/2)^n = \infty$ (the series diverges much faster). Log utility also diverges. Stronger resolutions needed.

Truncated St. Petersburg: cap the payoff at \M$. With reasonable$M$(say$M = $10^9$), the expected payoff is$\log_2(M)$— about$$30$. People still won’t pay that much, suggesting the cap explanation isn’t the whole story.

Probability-weighted St. Petersburg: change the coin probability. With probability $p$ for “continue” and $1-p$ for “stop and pay $2^n$”, the expected value diverges for $p \geq 1/2$ (which includes the original $p = 1/2$). For $p < 1/2$, expected value is finite.

These variants have been studied extensively in the academic literature on decision theory.

What the paradox teaches

The deepest lesson of the St. Petersburg paradox is that expected value alone is not sufficient for rational decision-making. The standard “maximize expected outcome” rule fails dramatically in cases with heavy-tailed distributions. To capture how rational humans actually behave — and how rational humans should behave, when stakes matter — you need something more: utility functions, risk aversion, bounded resources, or behavioral corrections.

This is one of the foundational insights of modern decision theory. It applies far beyond the original coin-flipping game:

• Insurance makes sense even though premiums exceed expected payouts, because the utility of avoiding ruin is much higher than the utility of an equivalent expected payoff.
• Lotteries are bad investments by expected value but rational under prospect theory (people overweight small probabilities of large gains).
• Diversification in investment makes sense because of diminishing marginal utility, not because it changes expected returns.
• Charitable giving can be rational even though it reduces expected wealth, because money has decreasing marginal utility — giving \100$ to someone in poverty produces more utility than keeping it would.

For students, the St. Petersburg paradox is one of the cleanest demonstrations that intuition and mathematics can disagree, and that resolving the disagreement requires expanding the mathematics — not just admitting that intuition is wrong. The right response to a paradox is rarely “intuition is wrong” or “math is wrong.” It’s usually “the model is incomplete.”

For Daniel Bernoulli, in 1738, the paradox prompted what may be the first major application of a logarithm to a social-science question. The answer he found — diminishing marginal utility — is so central to modern economics that almost no one remembers it had to be discovered. The math was hidden in the casino’s offering. Bernoulli pulled it out.

Three centuries later, his answer is still the textbook one. Expected value is the right tool for some questions. For others — particularly when stakes are heavy-tailed — you need utility. The St. Petersburg paradox is what made that distinction permanent in mathematical economics.

Frequently asked

Why is it called the St. Petersburg paradox?

Because Daniel Bernoulli published his famous analysis of the problem in Commentarii of the Imperial Academy of Sciences in St. Petersburg in 1738. The original problem was posed by his cousin Nicholas Bernoulli around 1713. Despite the name, the paradox has nothing to do with the city specifically — it's about expected value, utility, and how humans actually make decisions.

Is the expected value really infinite?

Mathematically yes. The expected payoff is (1/2)·$2 + (1/4)·$4 + (1/8)·$8 + ... = $1 + $1 + $1 + ... = ∞. Each term contributes exactly $1, and the sum diverges. So in the standard expected-value calculation, you should pay any finite amount to play. This is what makes the situation paradoxical.

Has this been tested experimentally?

Yes. In experimental economics studies dating from the 1990s, most people refuse to pay more than $5–$25 to play, regardless of how the prize structure is framed. Bernoulli's prediction (log utility) gives roughly $4 for a wealth of $1000 — close to what people actually pay. Other utility functions give different predictions but they all match human behavior much better than infinite expected value.

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