In decision theory

Decision theory

Decision theory in economics, psychology, philosophy, mathematics, and statistics is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision...

, a Choquet integral is a way of measuring the expected utility of an uncertain event. It is applied specifically to membership functions

Membership function (mathematics)

The membership function of a fuzzy set is a generalization of the indicator function in classical sets. In fuzzy logic, it represents the degree of truth as an extension of valuation. Degrees of truth are often confused with probabilities, although they are conceptually distinct, because fuzzy...

 and capacities

Capacity of a set

In mathematics, the capacity of a set in Euclidean space is a measure of that set's "size". Unlike, say, Lebesgue measure, which measures a set's volume or physical extent, capacity is a mathematical analogue of a set's ability to hold electrical charge. More precisely, it is the capacitance of...

. In imprecise probability theory

Imprecise probability

Imprecise probability generalizes probability theory to allow for partial probability specifications, and is applicable when information is scarce, vague, or conflicting, in which case a unique probability distribution may be hard to identify...

, the Choquet integral is also used to calculate the lower expectation induced by a 2-monotone lower probability

Upper and lower probabilities

Upper and lower probabilities are representations of imprecise probability. Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, this method uses two numbers: the upper probability of the event and the lower probability of the event.Because...

, or the upper expectation induced by a 2-alternating upper probability

Upper and lower probabilities

Upper and lower probabilities are representations of imprecise probability. Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, this method uses two numbers: the upper probability of the event and the lower probability of the event.Because...

. This integral was created by the French mathematician Gustave Choquet

Gustave Choquet

Gustave Choquet was a French mathematician.Choquet was born in Solesmes, Nord. His contributions include work in functional analysis, potential theory, topology and measure theory...

.

Using the Choquet integral to denote the expected utility of belief functions measured with capacities is a way to reconcile the Ellsberg paradox

Ellsberg paradox

The Ellsberg paradox is a paradox in decision theory and experimental economics in which people's choices violate the expected utility hypothesis.An alternate viewpoint is that expected utility theory does not properly describe actual human choices...

 and the Allais paradox

Allais paradox

The Allais paradox is a choice problem designed by Maurice Allais to show an inconsistency of actual observed choices with the predictions of expected utility theory.-Statement of the Problem:...

.

Definition

More specifically, let be a set, and let be any collection of subsets of . Consider a function and a monotone set function

Set function

In mathematics, a set function is a function whose input is a set. The output is usually a number. Often the input is a set of real numbers, a set of points in Euclidean space, or a set of points in some measure space.- Examples :...

 .

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

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. The Riemann integral is unsuitable for many theoretical purposes...

 (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

Homogeneous function

In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...

For all it holds that

Comonotone additivity

Additive function

In mathematics the term additive function has two different definitions, depending on the specific field of application.In algebra an additive function is a function that preserves the addition operation:for any two elements x and y in the domain. For example, any linear map is additive...

If are comonotone functions, that is, if for all it holds that.
then

Subadditivity

If is 2-alternating, then

Superadditivity

If is 2-monotone, then

Alternative Representation

Let denote a cumulative distribution function

Cumulative distribution function

In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...

 such that is integrable. Then this following formula is often referred to as Choquet Integral:
where .

• choose to get ,
• choose to get

External links

• Gilboa I., Schmeidler D.
  David Schmeidler
  David Schmeidler is an Israeli mathematician and economic theorist with important contributions in the theory of individual decision making under uncertainty . He currently holds professorships at Ohio State University and Tel Aviv University. He received his PhD in Mathematics from the Hebrew...
  
  (1992), Additive Representations of Non-Additive Measures and the Choquet Integral, Discussion Paper n° 985...