John von Neumann and Oskar Morgenstern's 1944 Theory of Games and Economic Behavior proved that if an agent's preferences over lotteries satisfy four axioms, then there exists a utility function such that the agent prefers the lottery with the higher expected utility. This representation theorem is one of the most important results in decision theory.
The Four Axioms
2. Transitivity: A ≽ B, B ≽ C → A ≽ C
3. Continuity: A ≻ B ≻ C → ∃p: pA + (1−p)C ~ B
4. Independence: A ≽ B → pA + (1−p)C ≽ pB + (1−p)C
Independence is the most controversial axiom. It states that mixing two lotteries with a common third lottery does not change their relative ranking. Allais (1953) constructed compelling counterexamples showing that intelligent decision makers systematically violate independence, launching the research program that eventually produced prospect theory and other non-expected utility models.
The Representation Theorem
If the four axioms hold, there exists a function u: outcomes → ℝ such that L₁ ≽ L₂ ⟺ EU(L₁) ≥ EU(L₂). The utility function is unique up to positive affine transformation (interval scale), meaning that the zero point and unit are arbitrary but ratios of utility differences are meaningful.