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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 ....

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