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| {{Refimprove|date=January 2011}}In [[decision theory]], a '''Choquet integral''' is a way of measuring the expected utility of an uncertain event. It is applied specifically to [[membership function (mathematics)|membership functions]] and [[Capacity of a set|capacities]]. In [[Imprecise probability|imprecise probability theory]], the Choquet integral is also used to calculate the lower expectation induced by a 2-monotone [[Upper and lower probabilities|lower probability]], or the upper expectation induced by a 2-alternating [[Upper and lower probabilities|upper probability]]. This integral was created by the French mathematician [[Gustave Choquet]].
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| Using the Choquet integral to denote the expected utility of belief functions measured with capacities is a way to reconcile the [[Ellsberg paradox]] and the [[Allais paradox]].<ref>Chateauneuf A., Cohen M. D., [http://hal-paris1.archives-ouvertes.fr/docs/00/34/88/22/PDF/V08087.pdf "Cardinal extensions of EU model based on the Choquet integral"], Document de Travail du Centre d’Economie de la Sorbonne n° 2008.87</ref>
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| ==Definition==
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| More specifically, let <math>S</math> be a set, and let <math>\mathcal{F}</math> be any collection of subsets of <math>S</math>. Consider a function <math>f : S\to \mathbb{R}</math> and a monotone [[set function]] <math>\nu : \mathcal{F}\to \mathbb{R}^+</math>.
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| Assume that <math>f</math> is measurable with respect to <math>\nu</math>, that is
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| :<math>\forall x\in\mathbb{R}\colon \{s | f (s) \geq x\}\in\mathcal{F}</math>
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| Then the Choquet integral of <math>f</math> with respect to <math>\nu</math> is defined by:
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| :<math>
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| (C)\int f d\nu :=
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| \int_{-\infty}^0
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| (\nu (\{s | f (s) \geq x\})-\nu(S))\, dx
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| +
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| \int^\infty_0
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| \nu (\{s | f (s) \geq x\})\, dx
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| </math>
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| where the integrals on the right-hand side are the usual [[Riemann integral]] (the integrands are integrable because they are monotone in <math>x</math>).
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| ==Properties==
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| In general the Choquet integral does not satisfy additivity. More specifically, if <math>\nu</math> is not a probability measure, it may hold that
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| :<math>\int f \,d\nu + \int g \,d\nu \neq \int (f + g)\, d\nu.</math>
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| for some functions <math>f</math> and <math>g</math>. | |
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| The Choquet integral does satisfy the following properties.
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| ===Monotonicity===
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| If <math>f\leq g</math> then
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| :<math>(C)\int f\, d\nu \leq (C)\int g\, d\nu</math>
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| ===Positive homogeneity===
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| For all <math>\lambda\ge 0</math> it holds that
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| :<math>(C)\int \lambda f \,d\nu = \lambda (C)\int f\, d\nu,</math>
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| ===Comonotone additivity===
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| If <math>f,g : S \rightarrow \mathbb{R}</math> are comonotone functions, that is, if for all <math>s,s' \in S</math> it holds that
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| :<math>(f(s) - f(s')) (g(s) - g(s')) \geq 0</math>.
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| then
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| :<math>(C)\int\, f d\nu + (C)\int g\, d\nu = (C)\int (f + g)\, d\nu.</math>
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| ===Subadditivity===
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| If <math>\nu</math> is 2-alternating,{{clarify|reason=What does 2-alternating mean?|date=July 2012}} then
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| :<math>(C)\int\, f d\nu + (C)\int g\, d\nu \ge (C)\int (f + g)\, d\nu.</math>
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| ===Superadditivity===
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| If <math>\nu</math> is 2-monotone,{{clarify|reason=What does 2-monotone mean?|date=July 2012}} then
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| :<math>(C)\int\, f d\nu + (C)\int g\, d\nu \le (C)\int (f + g)\, d\nu.</math>
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| ==Alternative Representation==
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| Let <math>G</math> denote a [[cumulative distribution function]] such that <math>G^{-1}</math> is <math>d H</math> integrable. Then this following formula is often referred to as Choquet Integral:
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| :<math>\int_{-\infty}^\infty G^{-1}(\alpha) d H(\alpha) = -\int_{-\infty}^a H(G(x))dx+ \int_a^\infty \hat{H}(1-G(x)) dx,</math>
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| where <math>\hat{H}(x)=H(1)-H(1-x)</math>.
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| * choose <math>H(x):=x</math> to get <math>\int_0^1 G^{-1}(x)dx = E[X]</math>,
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| * choose <math>H(x):=1_{[\alpha,x]}</math> to get <math>\int_0^1 G^{-1}(x)dH(x)= G^{-1}(\alpha)</math>
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| == See also ==
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| * [[Nonlinear expectation]]
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| * [[Superadditivity]]
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| * [[Subadditivity]]
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| ==Notes==
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| {{Reflist}}
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| == External links ==
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| *Gilboa I., [[David Schmeidler|Schmeidler D.]] (1992), [https://europealumni.kellogg.northwestern.edu/research/math/papers/985.pdf Additive Representations of Non-Additive Measures and the Choquet Integral], Discussion Paper n° 985...
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| [[Category:Decision theory]]
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| [[Category:Functional analysis]]
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Alyson is what my spouse loves to contact me but I don't like when people use my full name. Her family lives in Alaska but her husband desires them to transfer. One of the things she enjoys most is canoeing and she's been doing it for quite a while. Office supervising is my occupation.
My page ... tarot card readings