Choquet integral

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A Choquet integral is a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953.[1] It was initially used in statistical mechanics and potential theory,[2] but found its way into decision theory in the 1980s,[3] 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.

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.[4][5]


Definition[edit]

More specifically, let S be a set, and let \mathcal{F} be any collection of subsets of S. Consider a function f : S\to \mathbb{R} and a monotone set function \nu : \mathcal{F}\to \mathbb{R}^+.

Assume that f is measurable with respect to \nu, that is

\forall x\in\mathbb{R}\colon \{s | f (s) \geq x\}\in\mathcal{F}

Then the Choquet integral of f with respect to \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).

Properties[edit]

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

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

for some functions f and g.

The Choquet integral does satisfy the following properties.

Monotonicity[edit]

If f\leq g then

(C)\int f\, d\nu \leq (C)\int g\, d\nu

Positive homogeneity[edit]

For all \lambda\ge 0 it holds that

(C)\int \lambda f \,d\nu = \lambda (C)\int f\, d\nu,

Comonotone additivity[edit]

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

(f(s) - f(s')) (g(s) - g(s')) \geq 0.

then

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

Subadditivity[edit]

If \nu is 2-alternating,[clarification needed] then

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

Superadditivity[edit]

If \nu is 2-monotone,[clarification needed] then

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

Alternative Representation[edit]

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:

\int_{-\infty}^\infty G^{-1}(\alpha) d H(\alpha) = -\int_{-\infty}^a H(G(x))dx+ \int_a^\infty \hat{H}(1-G(x)) dx,

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

  • choose H(x):=x to get \int_0^1 G^{-1}(x)dx = E[X],
  • choose H(x):=1_{[\alpha,x]} to get \int_0^1 G^{-1}(x)dH(x)= G^{-1}(\alpha)

See also[edit]

Notes[edit]

  1. ^ Choquet, G. (1953). "Theory of capacities". Ann. Inst. Fourier, Grenoble 5: 131–295. 
  2. ^ Denneberg, D. (1994). Non-additive measure and Integral. Kluwer Academic. 
  3. ^ 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. 
  4. ^ Chateauneuf, A.; Cohen, M. D. (2010). "Cardinal Extensions of the EU Model Based on the Choquet Integral". In Bouyssou, Denis; Dubois, Didier; Pirlot, Marc et al. Decision-making Process: Concepts and Methods. doi:10.1002/9780470611876.ch10. 
  5. ^ Sriboonchita, S., Wong, W. K., Dhompongsa, S., & Nguyen, H. T. (2010). Stochastic dominance and applications to finance, risk and economics. CRC Press.

Further reading[edit]