Von Neumann–Morgenstern utility theorem

The theorem states that preferences over lotteries satisfying completeness, transitivity, continuity, and independence can be represented by maximizing expected utility. Its utility function is unique up to positive affine transformations.

Lottery: A probability distribution over possible outcomes. The theorem represents preferences over these probability distributions, not merely over certain outcomes.
Independence axiom: An axiom requiring preference between two lotteries to persist when both are mixed with the same third lottery in the same proportions. This mixture condition drives the linearity of utility over probabilities.
John von Neumann: A Hungarian-American mathematician and physicist who helped establish game theory and modern computing. He coauthored the axiomatic treatment of utility that gives the theorem its name.
Allais paradox: A pair of choices whose observed pattern violates the independence axiom of expected utility. It demonstrates that plausible preferences can fail a central theorem assumption.
Preference relation: A relation that ranks alternatives by preference or indifference. The theorem begins with a ranking of lotteries and imposes axioms on that ranking.
Continuity axiom (decision theory): An axiom requiring intermediate lotteries to be comparable to suitable mixtures of a preferred and a less-preferred outcome. Continuity supplies the probability mixtures needed to assign numerical utility levels.
Oskar Morgenstern: An Austrian-American economist known for foundational work in game theory and economic methodology. He coauthored the work that presented the expected-utility representation.
Prospect theory: A model of decisions under risk in which people evaluate gains and losses relative to a reference point. It explains systematic departures from expected utility, including responses to certainty and loss.
Expected value: The probability-weighted average of the values associated with possible outcomes. Expected utility applies this averaging operation to utility, rather than directly to outcomes.
Expected utility: The probability-weighted average of the utilities of a lottery’s possible outcomes. The theorem’s conclusion represents each lottery’s ranking by this quantity.

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Expected utility hypothesisBroader topic: Its axioms provide the formal foundation for the hypothesis.