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
![{\displaystyle \forall x\in \mathbb {R} \colon \{s|f(s)\geq x\}\in {\mathcal {F}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b928c4c32a96783dcc947d76c8d6e16221af4e5)
Then the Choquet integral of
with respect to
is defined by:
![{\displaystyle (C)\int fd\nu :=\int _{-\infty }^{0}(\nu (\{s|f(s)\geq x\})-\nu (S))\,dx+\int _{0}^{\infty }\nu (\{s|f(s)\geq x\})\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb390313275674322fa491d1eb0b31efe6ec9fcd)
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
![{\displaystyle \int f\,d\nu +\int g\,d\nu \neq \int (f+g)\,d\nu .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/994687ae2b2db78f14dea05141e75bf89234dcf6)
for some functions
and
.
The Choquet integral does satisfy the following properties.
If
then
![{\displaystyle (C)\int f\,d\nu \leq (C)\int g\,d\nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8440b361d1fca5ad2f1f15ec4bd35a5728bd7452)
For all
it holds that
![{\displaystyle (C)\int \lambda f\,d\nu =\lambda (C)\int f\,d\nu ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e75730942a8e85e4ee510ae08377441c630e528)
If
are comonotone functions, that is, if for all
it holds that
.
then
![{\displaystyle (C)\int \,fd\nu +(C)\int g\,d\nu =(C)\int (f+g)\,d\nu .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0114132452f73902b55f1c3566f322070a065ec4)
If
is 2-alternating, then
![{\displaystyle (C)\int \,fd\nu +(C)\int g\,d\nu \geq (C)\int (f+g)\,d\nu .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00c1fadf884e6c5096adc97b34e14a2045383744)
If
is 2-monotone, then
![{\displaystyle (C)\int \,fd\nu +(C)\int g\,d\nu \leq (C)\int (f+g)\,d\nu .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/994086782b3880a6d6b4c73e7f6dd96e477bf84e)
Alternative Representation
Let
denote a cumulative distribution function such that
is
integrable. Then this following formula is often referred to as Choquet Integral:
![{\displaystyle \int _{-\infty }^{\infty }G^{-1}(\alpha )dH(\alpha )=-\int _{-\infty }^{a}H(G(x))dx+\int _{a}^{\infty }{\hat {H}}(1-G(x))dx,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85ca6540d84fb1ab7e0429f7b2246a0bd7ab82b4)
where
.
- choose
to get
,
- choose
to get ![{\displaystyle \int _{0}^{1}G^{-1}(x)dH(x)=G^{-1}(\alpha )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4a8613717c73d7694c516184a738429bff351a9)
Notes
External links