Von NeumannMorganstern Expected Utility TheoryIn 1944, John Von Neumann and Oskar Morgenstern published their book, Theory of Games and Economic Behavior. In this book, they moved on from Bernoulli's formulation of a utlity function over wealth, and defined an expected utility function over lotteries, or gambles. Theirs is an axiomatic derivation, meaning, a set of assumptions over people's preferences is required before one can construct a utility function. Defining Lotteriesvon Neumann and Morgenstern weren't exactly referring to Powerball when they spoke of lotteries (although Powerball is one of many kinds of gambles that the theory describes). In their definition, a lottery or gamble is simply a probability distribution over a known, finite set of outcomes. These outcomes could be anything  amounts of money, goods, or even events. In this framework, we know for certain what the probability of the occurrence of each outcome is. Getting back to our earlier examples, we can frame them as:
Each of these outcomes had a probability attached to it, and so we can define a simple lottery as a set of outcomes, A={a_{1}, a_{2},...,a_{n}} each of which occurs with some known probability p_{i}. The Preference AxiomsIn order to construct a utility function over lotteries, or gambles, we will make the following assumptions on people's preferences. We denote the binary preference relation "is weakly preferred to" by , which includes both "strictly preferred to", and "indifferent to".
The last axiom allows us to reduce compound lotteries to simple lotteries, since one can also be similarly indifferent between a a simple lottery giving an outcome x with a probability p, and compound lottery where the prize might be yet another lottery ticket, allowing one to participate in a lottery with x as a possible outcome, such that the effective probability of getting x was p. The Expected Utility PropertyA utility function u is said to have the expected utility property if, for a gamble g with outcomes {a_{1}, a_{2},...,a_{n}}, with effective probabilities p_{1}, p_{2},...,p_{n} respectively, we have: u(g) = p_{1}u(a_{1}) + p_{2}u(a_{2}) + ... + p_{n}u(a_{n}) von Neumann and Morgenstern proved that, as long as all the preference axioms hold, then a utility function exists, and it satisfies the expected utility property. See here for a complete proof of the Expected Utility Theorem. As it stands, expected utility theory is widely used in theoretical and practical analysis  see the section on Applications of Expected Utility Theory for a detailed discussion. However, it is not without its flaws; in particular, many experiments have shown that people routinely violate the behavioral axioms. These have now become famous paradoxes in themselves  see the experimental discussion of the Allais Paradox and the Ellsberg Paradox. A Note on Utility FunctionsBy convention, we use the term Bernoulli Utility Function to refer to a decisionmaker's utility over wealth  since of course it was Bernoulli who originally proposed the idea that people's internal, subjective value for an amount of money was not necessarily equal to the physical value of that money. The term von NeumannMorgenstern Utility Function, or Expected Utility Function is used to refer to a decisionmaker's utility over lotteries, or gambles.  
