VNM Theorem

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

The VNM theorem is one of the classic results of Bayesian decision theory. It establishes that, under four assumptions,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 countable 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 B">A>B. We also have indifference, written A∼B. If A is either preferred to B or indifferent with B, this can be written A≥B.

The four VNM axioms are:

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

The VNM theorem is one of the classic results of Bayesian decision theory. It establishes that, under four assumptions, 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.)