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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.)

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  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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