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 (L_{n}), each one defined by possible outcomes (C_{1},C_{2},…,C_{n}) and their corresponding probabilities (p_{1}, p_{2},…,p_{i}, with ∑p_{i}=1).

*EU(L) = U(c _{2})p_{1} + U(c_{2})p_{2} + … + U(c_{n})p_{n}*