Choquet integral: Difference between revisions
unref |
No edit summary |
||
Line 1: | Line 1: | ||
{{Unreferenced|date=January 2011}}In [[decision theory]], a '''Choquet integral'''<!--, named after [[Gustave Choquet]],--> is a way of measuring the expected utility of an uncertain event. It is applied specifically to [[membership function (mathematics)|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]]. |
{{Unreferenced|date=January 2011}}In [[decision theory]], a '''Choquet integral'''<!--, named after [[Gustave Choquet]],--> is a way of measuring the expected utility of an uncertain event. It is applied specifically to [[membership function (mathematics)|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]]. |
||
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]]. |
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>. |
||
==Definition== |
==Definition== |
||
Line 69: | Line 69: | ||
* choose <math>H(x):=x</math> to get <math>\int_0^1 G^{-1}(x)dx = E[X]</math>, |
* choose <math>H(x):=x</math> to get <math>\int_0^1 G^{-1}(x)dx = E[X]</math>, |
||
* choose <math>H(x):=1_{[\alpha,x]}</math> to get <math>\int_0^1 G^{-1}(x)dH(x)= G^{-1}(\alpha)</math> |
* choose <math>H(x):=1_{[\alpha,x]}</math> to get <math>\int_0^1 G^{-1}(x)dH(x)= G^{-1}(\alpha)</math> |
||
==Notes== |
|||
{{Reflist}} |
|||
== External links == |
|||
*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... |
|||
[[Category:Decision theory]] |
[[Category:Decision theory]] |
Revision as of 00:40, 27 February 2011
In decision theory, a Choquet integral is a way of measuring the expected utility of an uncertain event. It is applied specifically to capacities. In imprecise probability theory, the Choquet integral is also used to calculate the lower expectation induced by a 2-monotone lower probability, or the upper expectation induced by a 2-alternating upper probability. This integral was created by the French mathematician Gustave Choquet.
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[1].
Definition
More specifically, let be a set, and let be any collection of subsets of . Consider a function and a monotone set function .
Assume that is measurable with respect to , that is
Then the Choquet integral of with respect to is defined by:
where the integrals on the right-hand side are the usual Riemann integral (the integrands are integrable because they are monotone in ).
Properties
In general the Choquet integral does not satisfy additivity. More specifically, if is not a probability measure, it may hold that
for some functions and .
The Choquet integral does satisfy the following properties.
If then
Positive homogeneity
For all it holds that
Comonotone additivity
If are comonotone functions, that is, if for all it holds that
- .
then
If is 2-alternating, then
If is 2-monotone, then
Alternative Representation
Let denote a cumulative distribution function such that is integrable. Then this following formula is often referred to as Choquet Integral:
where .
- choose to get ,
- choose to get
Notes
- ^ Chateauneuf A., Cohen M. D., "Cardinal extensions of EU model based on the Choquet integral", Document de Travail du Centre d’Economie de la Sorbonne n° 2008.87
External links
- Gilboa I., Schmeidler D. (1992), Additive Representations of Non-Additive Measures and the Choquet Integral, Discussion Paper n° 985...