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.