Expected utility theory rests on a small set of axioms about rational preferences over lotteries. When these axioms are satisfied, von Neumann and Morgenstern (1944) proved that preferences can be represented by a utility function and that rational agents maximize expected utility. These axioms define the standard against which human decision making is evaluated.
The Axioms
2. Transitivity: A ≽ B and B ≽ C → A ≽ C
3. Continuity: if A ≻ B ≻ C, ∃p: pA + (1−p)C ~ B
4. Independence: A ≽ B → pA + (1−p)C ≽ pB + (1−p)C
Completeness ensures every pair of options can be compared. Transitivity prevents cyclic preferences. Continuity (or the Archimedean axiom) ensures no outcome is infinitely good or bad. Independence is the most controversial: it states that mixing two options with a common third option does not change their relative ranking.
Violations and Alternatives
Systematic violations of independence were demonstrated by the Allais paradox (1953) and have motivated alternative theories. Prospect theory replaces independence with rank-dependent probability weighting. Regret theory drops transitivity. Rank-dependent utility theory satisfies all axioms except independence in its strong form. Each alternative theory corresponds to weakening or replacing specific axioms in the expected utility framework.