The St. Petersburg Paradox and Utility
Daniel Bernoulli's resolution of a paradox in expected-value theory — and the birth of the idea that money's value is not the same as its amount.
Bernstein covers a puzzle that exposed a genuine flaw in early probability theory's simplest form: the St. Petersburg paradox, involving a coin-flip game whose payout doubles each time the coin comes up heads in a row, which has a mathematically infinite expected value — yet virtually no one, when actually offered the game, would pay more than a modest amount to play it. This directly contradicted the assumption that a rational person should value a bet purely by its mathematical expected value, the same expected-value logic underlying the Pascal-Fermat breakthrough from earlier in this course.
Daniel Bernoulli (Jacob's nephew) resolved the paradox with a genuinely new idea: people do not value money linearly — an additional dollar matters less to someone who already has a great deal of money than to someone with very little, a concept now called diminishing marginal utility. Valuing the St. Petersburg game by its expected *utility* rather than its expected raw monetary value resolves the paradox and produces a finite, sensible price — and this insight, that risk should be evaluated relative to a person's specific circumstances and not by monetary value in the abstract, became foundational to how economists eventually formalized concepts of risk aversion.
Daniel Bernoulli's insight matters for this course's later chapters because it is the direct conceptual ancestor of modern risk aversion in economics and finance: if an additional dollar of gain is worth less than an additional dollar of loss hurts (a direct consequence of diminishing marginal utility applied symmetrically around a person's current wealth), a rational person should reasonably prefer a certain, smaller amount over a risky bet with the identical expected monetary value — the mathematical foundation for why diversification and risk-adjusted position sizing (this Book Club's Kelly criterion, covered in A Man for All Markets) make sense even for a purely rational, self-interested decision-maker, not just a psychologically anxious one.
- The St. Petersburg paradox showed that a bet with infinite mathematical expected value is one almost nobody would pay much to play — exposing a flaw in pure expected-value reasoning.
- Daniel Bernoulli resolved it with diminishing marginal utility — an additional dollar matters less to someone who already has more of them.
- This became the conceptual foundation for risk aversion in economics — the reason a rational person reasonably prefers a certain smaller amount over an equal-expected-value risky bet.