St. Petersburg paradox

A lottery pays an amount that doubles with each successive coin toss until the first heads, giving it infinite expected monetary value despite modest apparent appeal.

Expected value: The probability-weighted average of the possible outcomes of a random variable. Its sum over the lottery’s possible payouts diverges, producing infinite expected monetary value.
Daniel Bernoulli: A Swiss mathematician who developed an influential account of utility and risk in the eighteenth century. His 1738 treatment introduced diminishing marginal utility to explain the lottery’s limited appeal.
Pascal's wager: Blaise Pascal’s argument that finite costs may be outweighed by the prospect of infinite gain or loss. It shares the tension between infinite stakes and finite human choices, but concerns religious belief.
Marginal utility: The additional utility gained from consuming one more unit of a good. Bernoulli’s explanation relies on each extra unit of wealth adding less utility than the previous one.
Geometric distribution: A discrete probability distribution describing the number of independent trials required for the first success. It models how many coin tosses occur before the lottery pays.
Nicolas Bernoulli: A Swiss mathematician who posed the lottery problem that became known as the St. Petersburg paradox. His correspondence with Pierre Rémond de Montmort presented the original puzzle.
Kelly criterion: A formula for choosing the fraction of wealth to wager to maximize long-run logarithmic growth. It uses logarithmic utility in repeated betting rather than pricing one isolated lottery.
Prospect theory: A descriptive theory of choice under risk that models value relative to a reference point and weights probabilities nonlinearly. It offers a later psychological account of risky choices beyond expected utility.
Utility function: A function assigning numerical values to outcomes to represent preferences or decision-making priorities. Applying diminishing utility to money can make the lottery’s expected utility finite.
Pierre Rémond de Montmort: A French mathematician known for early work on probability and games of chance. Nicolas Bernoulli sent him the paradox in a letter, helping launch its mathematical discussion.

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