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.