Utility

: This article is about "utility" in economics and in game theory. For utility companies and similar concepts, see public utility. For utilities in computers, see computer software. See also Utility (patent)

Expected utility

A von Neumann-Morgenstern utility function u : X ightarrow extbf{R} assigns a real number to every element of the outcome space in a way that captures the agent's preferences over both simple and compound lotteries (put in category-theoretic language, u induces a morphism between the category of preferences under uncertainty and the category of reals). The agent will prefer a lottery L_1 to a lottery L_2 if and only if the expected utility (iterated over compound lotteries if necessary) of L_1 is greater than the expected utility of L_2.

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Restricting to the discrete choice context, let L : X ightarrow be a simple lottery such that L(x_i) = p_i, where p_i is the probability that x_i is won. We may also consider compound lotteries, where the prizes are themselves simple lotteries.

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The expected utility theorem says that a von Neumann-Morgenstern utility function exists if and only if the agent's preference relation >= on the space of simple lotteries satisfies four axioms: completeness, transitivity, convexity/continuity (also called the Archimedean property), and independence.

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Completeness and transitivity are discussed supra. The Archimedean property says that for simple lotteries L_1 >= L_2 >= L_3, then there exists a 0

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Independence is probably the most controversial of the axioms.

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Daniel Bernoulli has shown how the personal utility vary with the personal degree of risk aversion, itself linked to the initial wealth situation of the person.

Related Topics:
Daniel Bernoulli - Risk aversion

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