In 1931, mathematical philosopher Frank Ramsey pioneered the idea of subjective probability as a representation of an individual’s beliefs or uncertainties. Then, in the 1940s, mathematician John von Neumann and economist Oskar Morgenstern developed an axiomatic basis for utility theory as a way of expressing an individual’s preferences over uncertain outcomes. (This is in contrast to social-choice theory, which addresses the problem of deriving group preferences from individual preferences.) Statistician Leonard Jimmie Savage then developed an alternate axiomatic framework for decision analysis in the early 1950s. The resulting expected-utility theory provides a complete axiomatic basis for decision making under uncertainty.