The **VNM theorem** is one of the classic results of Bayesian decision theory. It establishes that, under four assumptions known as the **VNM axioms**, a preference relation *must* be representable by maximum-expectation decision making over some real-valued utility function. (In other words, rational decision making is best-average-case decision making.)

Starting with some set of outcomes, **gambles **(or **lotteries**) are defined recursively. An outcome is a gamble, and for any finite set of gambles, a probability distribution over those gambles is a gamble.

Preferences are then expressed over gambles via a preference relation. if is preferred to , this is written . We also have indifference, written . If is either preferred to *or* indifferent with , this can be written .

The four VNM axioms are:

**Completeness.**For any gambles and , either , , or .**Transitivity.**If and , then .**Continuity.**If , then there exists a probability such that . In other words, there is a probability which hits any point between two gambles.**Independence.**For any and , we have if and only if . In other words, substituting for in any gamble can’t make that gamble worth less.

In contrast to Utility Functions, this tag focuses *specifically* on posts which discuss the VNM theorem itself.