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

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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:

**Completeness.**For any gambles A and B, either B">A>B, A">B>A, or A∼B.**Transitivity.**If A<B and B<C, then A<C.**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.**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.

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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.)

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Created by abramdemski at 1y