A Choquet integral is a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953. It was initially used in statistical mechanics and potential theory, but found its way into decision theory in the 1980s, where it is used as a way of measuring the expected utility 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.
The following notation is used:
- - a set.
- - a collection of subsets of .
- - a function.
- - 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 ).
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.
For all it holds that
If are comonotone functions, that is, if for all it holds that
- which can be thought of as and rising and falling together
If is 2-alternating,[clarification needed] then
If is 2-monotone,[clarification needed] then
Let denote a cumulative distribution function such that is integrable. Then this following formula is often referred to as Choquet Integral:
- choose to get ,
- choose to get
The Choquet integral was applied in image processing, video processing and computer vision. In behavioral decision theory, Amos Tversky and Daniel Kahneman use the Choquet integral and related methods in their formulation of Cumulative Prospect Theory.
- Choquet, G. (1953). "Theory of capacities". Annales de l'Institut Fourier. 5: 131–295.
- Denneberg, D. (1994). Non-additive measure and Integral. Kluwer Academic.
- Grabisch, M. (1996). "The application of fuzzy integrals in multicriteria decision making". European Journal of Operational Research. 89 (3): 445–456. doi:10.1016/0377-2217(95)00176-X.
- Chateauneuf, A.; Cohen, M. D. (2010). "Cardinal Extensions of the EU Model Based on the Choquet Integral". In Bouyssou, Denis; Dubois, Didier; Pirlot, Marc; Prade, Henri (eds.). Decision-making Process: Concepts and Methods. doi:10.1002/9780470611876.ch10.
- Sriboonchita, S., Wong, W. K., Dhompongsa, S., & Nguyen, H. T. (2010). Stochastic dominance and applications to finance, risk and economics. CRC Press.
- Tversky, A.; Kahneman, D. (1992). "Advances in Prospect Theory: Cumulative Representation of Uncertainty". Journal of Risk and Uncertainty. 5: 297–323.