St. Petersburg Paradox

The St. Petersburg Paradox is a thought experiment in probability theory that exposes the inadequacy of expected monetary value as a criterion for rational decision-making. In the game, a fair coin is tossed until it lands heads; the player wins a prize that doubles with each toss, yielding an infinite expected value. Yet no rational person would pay more than a modest amount to play, revealing that human valuation is not linear in money. Daniel Bernoulli resolved the paradox by proposing that agents maximize expected utility rather than expected wealth, introducing the concept of diminishing marginal utility that underlies modern expected utility theory and risk aversion. The paradox is not merely a curiosity about gambling; it is the foundational crack in the edifice of classical economics that forced the separation of value from price.

The paradox also generates a deeper puzzle: if the expected value is infinite, why does the subjective valuation remain finite? Some have proposed that the resolution lies in the finitude of the casino's resources — a real casino cannot pay arbitrarily large sums. Others argue that the paradox reveals the limits of probability theory itself, suggesting that the concept of expectation requires constraints on the space of possible outcomes. The St. Petersburg Paradox is thus a boundary object between mathematics, psychology, and economics: it shows that no formal system can determine rational choice without importing assumptions about human psychology.