The expected utility theory deals with the analysis of 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 theory are lotteries, or gambles, (L_{n}) each one defined by all possible outcomes or consequences (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}*

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. 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:

1.Axioms of order:

Completeness L_{0}L_{1} or L_{1}L_{0 }or L_{0}∿L_{1}

Reflexive L_{0}L_{0}

Transitive L_{0}L_{1} ; L_{1}L_{2 }↔ L_{0}L_{2}

2.Continuity: L_{0}L_{1} ; L_{1}L_{2}

with p∊ [0,1] so that L_{1}∿ p L_{0}+(1-p)L_{2}

3.Rationality: in order to maximize results, the highest probability will be chosen (*ceteris paribus*)

4.Rational Equivalence: agents will evaluate rationally the probabilities of the different results

5.Independence (sure thing principle): if L_{0}L_{1 }→β*L_{0}+ (1- β)*L_{2}β*L_{1} +(1- β)*L_{2}