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In decision theory, the von Neumann-Morgenstern utility theorem shows that, under certain axioms of rational behavior, a decision-maker faced with risky (probabilistic) outcomes of different choices will behave as if he is maximizing the expected value of some function defined over the potential outcomes. This function is known as the von Neumann-Morgenstern utility function. The theorem is the basis for expected utility theory. In 1947, John von Neumann and Oskar Morgenstern proved that any individual whose preferences satisfied four 〔Neumann, John von and Morgenstern, Oskar, ''Theory of Games and Economic Behavior''. Princeton, NJ. Princeton University Press, 1953.〕 axioms has a utility function; such an individual's preferences can be represented on an interval scale and the individual will always prefer actions that maximize expected utility. That is, they proved that an agent is (VNM-)rational ''if and only if'' there exists a real-valued function ''u'' defined by possible outcomes such that every preference of the agent is characterized by maximizing the expected value of ''u'', which can then be defined as the agent's ''VNM-utility'' (it is unique up to adding a constant and multiplying by a positive scalar). No claim is made that the agent has a "conscious desire" to maximize ''u'', only that ''u'' exists. Any individual whose preferences violate von Neumann and Morgenstern's axioms would agree to a Dutch book, which is a set of bets that necessarily leads to a loss. Therefore, it is arguable that any individual who violates the axioms is irrational. The expected utility hypothesis is that rationality can be modeled as maximizing an expected value, which given the theorem, can be summarized as "''rationality is VNM-rationality''". VNM-utility is a ''decision utility'' in that it is used to describe ''decision preferences''. It is related but not equivalent to so-called ''E-utilities''〔Kahneman, Wakker and Sarin, 1997, ''Back to Bentham? Explorations of experienced utility'', The quarterly journal of economics.〕 (experience utilities), notions of utility intended to measure happiness such as that of Bentham's Greatest Happiness Principle. == Set-up == In the theorem, an individual agent is faced with options called ''lotteries''. Given some mutually exclusive outcomes, a lottery is a scenario where each outcome will happen with a given probability, all probabilities summing to one. For example, for two outcomes ''A'' and ''B'', :: denotes a scenario where ''P''(''A'') = 25% is the probability of ''A'' occurring and ''P''(''B'') = 75% (and exactly one of them will occur). More generally, for a lottery with many possible outcomes ''Ai'', we write :: with the sum of the s equalling 1. The outcomes in a lottery can themselves be lotteries between other outcomes, and the expanded expression is considered an equivalent lottery: 0.5(0.5''A'' + 0.5''B'') + 0.5''C'' = 0.25''A'' + 0.25''B'' + 0.50''C''. If lottery ''M'' is preferred over lottery ''L'', we write If ''M'' is either preferred over or viewed with indifference relative to ''L'', we write If the agent is indifferent between ''L'' and ''M'', we have the ''indifference relation''〔Kreps, David M. ''Notes on the Theory of Choice''. Westview Press (May 12, 1988), chapters 2 and 5.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Von Neumann–Morgenstern utility theorem」の詳細全文を読む スポンサード リンク
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