Choquet integral

In decision theory, a Choquet integral is 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. This integral was created by the French mathematician Gustave Choquet.

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.

Definition

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

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

If $f\leq g$ then

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

Positive homogeneity

For all $\forall\lambda\ge 0$ it holds that

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

Comonotone additivity

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

If $\nu$ is 2-alternating, then

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

Superadditivity

If $\nu$ is 2-monotone, then

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

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:

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

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 $\int_0^1 G^{-1}(x)dH(x)= G^{-1}(\alpha)$

External links

• Gilboa I., Schmeidler D. (1992), Additive Representations of Non-Additive Measures and the Choquet Integral, Discussion Paper n° 985...