VNM Theorem

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 $A\ge B$.

The four VNM axioms are:

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