Leonard Savage's 1954 The Foundations of Statistics extended expected utility theory from risk (known probabilities) to uncertainty (unknown probabilities). Savage's framework derives both a subjective probability measure and a utility function from axioms on preferences over acts — functions from states of the world to outcomes. This unified treatment of belief and value is one of the deepest results in decision theory.
Savage's Framework
P2: Sure-thing principle (conditional independence)
P3: State-independent preferences over outcomes
P4: Qualitative probability from preferences
P5: Non-triviality
P6–P7: Continuity and dominance conditions
The sure-thing principle (P2) is the analog of the independence axiom: if two acts have the same consequence in some states, then preference between them should depend only on the states where they differ. This axiom has been challenged by the Ellsberg paradox, which shows that people are averse to ambiguity (unknown probabilities) in ways that violate P2.
Significance
Savage's theorem shows that rational behavior under uncertainty is as if the agent had a probability distribution over states and maximized expected utility. This provides the theoretical foundation for Bayesian decision theory, Bayesian statistics, and rational models of cognition. The framework remains the gold standard for normative decision theory.