The expected utility theory deals with the analysis of choices among risky projects with multiple possible outcomes.

The expected utility (EU) hypothesis is a popular concept in economics, game theory and decision theory that serves as a reference guide for judging decisions involving uncertainty. The theory recommends which option a rational individual should choose in a complex situation, based on his tolerance for risk and personal preferences. The EU of an agent's risky decision is the mathematical expectation of his utility from different outcomes given their probabilities.

Although the EU hypothesis is standard in economic modelling, largely because of its simplicity and convenience, it has been found to be violated in psychology experiments. For many years, psychologists and economic theorists have been developing new theories to explain these deficiencies. These include prospect theory, rank-dependent EU and cumulative prospect theory.

Sources: Wikipedia 1 for the excerpt, Wikipedia 2 for the body.