Choquet integral

 ( Expected Utility ) 

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 theory 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 and the Allais paradox.

(2025). 9781420082661, CRC Press. ISBN 9781420082661

Multiobjective optimization problems seek Pareto optimal solutions, but the Pareto set of such solutions can be extremely large, especially with multiple objectives. To manage this, optimization often focuses on a specific function, such as a weighted sum, which typically results in solutions forming a convex envelope of the feasible set. However, to capture non-convex solutions, alternative aggregation operators like the Choquet integral can be used.Lust, Thibaut & Rolland, Antoine. (2014). 2-additive Choquet Optimal Solutions in Multiobjective Optimization Problems. Communications in Computer and Information Science. 442. 256-265. 10.1007/978-3-319-08795-5_27.

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Definition The following notation is used:

• $S$ – a set.
• $\backslash mathcal\{F\}$ – a collection of subsets of $S$.
• $f\; :\; S\backslash to\; \backslash mathbb\{R\}$ – a function.
• $\backslash nu\; :\; \backslash mathcal\{F\}\backslash to\; \backslash mathbb\{R\}^+$ – a monotone set function.

Assume that $f$ is measurable with respect to $\backslash mathcal\{F\}$, that is

$\backslash forall\; x\backslash in\backslash mathbb\{R\}\backslash colon\; \backslash \{s\; \backslash in\; S\; \backslash mid\; f\; (s)\; \backslash geq\; x\backslash \}\backslash in\backslash mathcal\{F\}$

Then the Choquet integral of $f$ with respect to $\backslash nu$ is defined by:

$$

(C)\int f d\nu := \int_{-\infty}^0 (\nu (\{s | f (s) \geq x\})-\nu(S))\, dx + \int^\infty_0 \nu (\{s | f (s) \geq x\})\, dx

where the integrals on the right-hand side are the usual Riemann integral (the integrands are integrable because they are monotone in $x$).

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Properties In general the Choquet integral does not satisfy additivity. More specifically, if $\backslash nu$ is not a probability measure, it may hold that

$\backslash int\; f\; \backslash ,d\backslash nu\; +\; \backslash int\; g\; \backslash ,d\backslash nu\; \backslash neq\; \backslash int\; (f\; +\; g)\backslash ,\; d\backslash nu.$

for some functions $f$ and $g$.

The Choquet integral does satisfy the following properties.

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Monotonicity If $f\backslash leq\; g$ then

$(C)\backslash int\; f\backslash ,\; d\backslash nu\; \backslash leq\; (C)\backslash int\; g\backslash ,\; d\backslash nu$

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Positive homogeneity For all $\backslash lambda\backslash ge\; 0$ it holds that

$(C)\backslash int\; \backslash lambda\; f\; \backslash ,d\backslash nu\; =\; \backslash lambda\; (C)\backslash int\; f\backslash ,\; d\backslash nu,$

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Comonotone additivity If $f,g\; :\; S\; \backslash rightarrow\; \backslash mathbb\{R\}$ are comonotone functions, that is, if for all $s,s\text{'}\; \backslash in\; S$ it holds that

$(f(s)\; -\; f(s\text{'}))\; (g(s)\; -\; g(s\text{'}))\; \backslash geq\; 0$.

which can be thought of as $f$ and $g$ rising and falling together

then

$(C)\backslash int\backslash ,\; f\; d\backslash nu\; +\; (C)\backslash int\; g\backslash ,\; d\backslash nu\; =\; (C)\backslash int\; (f\; +\; g)\backslash ,\; d\backslash nu.$

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Subadditivity If $\backslash nu$ is 2-alternating, then

$(C)\backslash int\backslash ,\; f\; d\backslash nu\; +\; (C)\backslash int\; g\backslash ,\; d\backslash nu\; \backslash ge\; (C)\backslash int\; (f\; +\; g)\backslash ,\; d\backslash nu.$

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Superadditivity If $\backslash nu$ is 2-monotone, then

$(C)\backslash int\backslash ,\; f\; d\backslash nu\; +\; (C)\backslash int\; g\backslash ,\; d\backslash nu\; \backslash le\; (C)\backslash int\; (f\; +\; g)\backslash ,\; d\backslash nu.$

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Alternative representation Let $G$ denote a cumulative distribution function such that $G^\{-1\}$ is $d\; H$ integrable. Then this following formula is often referred to as Choquet Integral:

$\backslash int\_\{-\backslash infty\}^\backslash infty\; G^\{-1\}(\backslash alpha)\; d\; H(\backslash alpha)\; =\; -\backslash int\_\{-\backslash infty\}^a\; H(G(x))dx+\; \backslash int\_a^\backslash infty\; \backslash hat\{H\}(1-G(x))\; dx,$

where $\backslash hat\{H\}(x)=H(1)-H(1-x)$.

• choose $H(x):=x$ to get $\backslash int\_0^1\; G^\{-1\}(x)dx\; =\; EX$,
• choose $H(x):=1\_\{\backslash alpha,x\}$ to get $\backslash int\_0^1\; G^\{-1\}(x)dH(x)=\; G^\{-1\}(\backslash alpha)$

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Applications 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.

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See also

• Nonlinear expectation
• Superadditivity
• Subadditivity

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Notes

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Further reading

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