A Choquet integral is a
subadditive In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. ...
or
superadditive integral created by the French mathematician
Gustave Choquet in 1953. It was initially used in
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
and
potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...
, but found its way into
decision theory
Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical ...
in the 1980s, where it is used as a way of measuring the expected
utility
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosoph ...
of an uncertain event. It is applied specifically to
membership functions and
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.
Using the Choquet integral to denote the expected utility of belief functions measured with capacities is a way to reconcile the
Ellsberg paradox
In decision theory, the Ellsberg paradox (or Ellsberg's paradox) is a paradox in which people's decisions are inconsistent with subjective expected utility theory. Daniel Ellsberg popularized the paradox in his 1961 paper, “Risk, Ambiguity, an ...
and the
Allais paradox
The Allais paradox is a choice problem designed by to show an inconsistency of actual observed choices with the predictions of expected utility theory.
Statement of the problem
The Allais paradox arises when comparing participants' choices in two ...
.
Definition
The following notation is used:
*
– a set.
*
– a collection of subsets of
.
*
– a function.
*
– a monotone
set function
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R an ...
.
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
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of GÃ ...
(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.
Monotonicity
If
then
:
Positive homogeneity
For all
it holds that
:
Comonotone additivity
If
are comonotone functions, that is, if for all
it holds that
:
.
:which can be thought of as
and
rising and falling together
then
:
Subadditivity
If
is 2-alternating, then
:
Superadditivity
If
is 2-monotone, then
:
Alternative representation
Let
denote a
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
such that
is
integrable. Then this following formula is often referred to as Choquet Integral:
:
where
.
* choose
to get