The term expected utility was first introduced by Daniel Bernoulli who used it to solve the St. Petersburg paradox, as the expected value was not sufficient for its resolution. He introduce the term in his paper “Commentarii Academiae Scientiarum Imperialis Petropolitanae” (translated as “Exposition of a new theory on the measurement of risk”), 1738, where he solved the paradox. However, John von Neumann and Oskar Morgenstern, in their book “Theory of Games and Economic Behavior”, 1944, considered the cornerstone of expected utility theory, provided great contributions and built a mathematical foundation for Bernoulli’s solution of the paradox. They developed a set of axioms for the preferential relations in order to guarantee that the utility function is well-behaved.
The expected utility is used to provide an answer to situations where individuals must make a decision without knowing which outcomes may result from that decision, this is, decision making under uncertainty. These individuals will choose the act that will result in the highest expected utility, being this the sum of the products of probability and utility over all possible outcomes. The decision made will also depend on the agent’s risk aversion and the utility of other agents.
The base of the expected utility theory are lotteries (Ln), each one defined by possible outcomes (C1,C2,…,Cn) and their corresponding probabilities (p1, p2,…,pi, with ∑pi=1).
EU(L) = U(c2)p1 + U(c2)p2 + … + U(cn)pn