Axiomatic utility theory under risk

The initial development of utility theory for decision-making under risk began with Cramer (1728) and Bernoulli (1738). Bernoulli's analysis of the St. Petersburg Paradox—a lottery with infinite expected monetary value—revealed that most individuals would not invest significantly in such a gamble. To explain this behavior, he suggested replacing expected monetary value with expected utility, or "moral expectation," as the key criterion for risky choices. However, Bernoulli's argument and his use of a logarithmic utility function seem somewhat arbitrary, relying heavily on intuitively appealing examples. It wasn't until two centuries later that von Neumann and Morgenstern (1947) demonstrated that if a decision maker's preferences meet specific conditions, they can be represented by the expected value of a real-valued utility function based on the consequences. Although both Bernoulli's and von Neumann and Morgenstern's theories share a similar mathematical framework for expected utility, they differ significantly in their foundational approaches. Cramer (1728) also contributed by proposing that the utility of a monetary amount is represented by its square root.