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 $A$ is preferred to $B$ , this is written $A > B$ . We also have indifference, written $A \sim B$ . If $A$ is either preferred to $B$ or indifferent with $B$ , this can be written $A \geq B$ .

The four VNM axioms are:

Completeness. For any gambles $A$ and $B$ , either $A > B$ , $B > A$ , or $A \sim B$ .
Transitivity. If $A < B$ and $B < C$ , then $A < C$ .
Continuity. If $A \leq B \leq C$ , then there exists a probability $p \in [0, 1]$ such that $p A + (1 - p) C \sim B$ . In other words, there is a probability which hits any point between two gambles.
Independence. For any $C$ and $p \in [0, 1]$ , we have $A \leq B$ if and only if $p A + (1 - p) C \leq p B + (1 - p) C$ . In other words, substituting $A$ for $B$ 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.