Abstract
Let \(\Omega \) be an arbitrary set, equipped with an algebra \({\mathcal {A}}\subseteq 2^{\Omega }\) and let \(f: B({\mathcal {A}}) \rightarrow {\mathbb {R}}\) be a functional defined on the set \(B({\mathcal {A}}) \) of bounded measurable functions \(x:\Omega \rightarrow {\mathbb {R}}\). We provide necessary and sufficient conditions for a submodular functional f to be representable as a Choquet integral. From standard properties of the Choquet integral the functional f should be positively homogeneous and constant additive. Our first result shows that these two properties, together with submodularity, characterize a subadditive Choquet integral, when \(\Omega \) is finite. In the general case, f is a subadditive Choquet integral if and only if it satisfies the three previous properties, together with sup-norm continuity. This result provides another characterization of subadditive Choquet integrals different from the seminal paper by Schmeidler (Proc Am Math Soc 97(2):255–261, 1986) that relies on comonotonic additivity.
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Notes
The functional f is said to be comonotonic additive if \(f(x+y)= f(x)+f(y)\) for all x, y in \(B({\mathcal {A}}) \) such that \(x+y\in B({\mathcal {A}}) \) and \((x(\omega )-x(\omega '))(y(\omega )-y(\omega '))\ge 0\) for all \(\omega ,\omega '\) in \(\Omega \).
When \(\Omega \) is finite of cardinal n, we can identify \({\mathbb {R}}^{\Omega }\) with \({\mathbb {R}}^{n}\), thus a function \(x:\Omega \rightarrow {\mathbb {R}}\) can also be viewed as the n-tuple \(x=(x_{1}, \dots , x_{n})\). The previously defined order is the coordinate-wise order of \({\mathbb {R}}^{n}\), i.e., \(x=(x_{1}, \dots , x_{n})\le y=(y_{1}, \dots , y_{n})\) in \({\mathbb {R}}^{n}\) means \(x_{i}\le y_{i}\) for every \(i=1,\dots ,n\). The lattice operations \(\wedge \) and \(\vee \) are defined by \(x\wedge y:= (\min \{x_{1},y_{1}\}), \dots , \min \{x_{n},y_{n}\}), x\vee y:= (\max \{x_{1},y_{1}\}), \dots , \max \{x_{n},y_{n}\})\). With the previous identification, for \(A\subseteq \{1,\dots , n\}, \varvec{1}_{A}\) will now be the vector in \({\mathbb {R}}^{n}\) such that \(x_{i}=1\) if \(i\in A\) and \(x_{i}=0\) otherwise. Thus we denote by \(\varvec{1}_{i}:= \varvec{1}_{\{i\}}\) (resp. \(\varvec{1}_{\Omega }\)) the vector with all coordinates equal to zero, but the i-th equal to 1 (resp. with all coordinates equal to 1) so that \(x=(x_{1}, \dots , x_{n})= x_{1}\varvec{1}_{1}+ \dots +x_{n}\varvec{1}_{n}\).
This is easily proved to be equivalent to \(f(x +t \varvec{1}_{\Omega })= f(x) +t f(\varvec{1}_{\Omega })\) for all \(t\in {\mathbb {R}}\), all \(x\in {\mathbb {R}}^{\Omega }\). Also, under positive homogeneity, constant additivity is equivalent to
$$\begin{aligned} f(x + \varvec{1}_{\Omega })= f(x) + f(\varvec{1}_{\Omega }) \hbox { for all } x\in {\mathbb {R}}^{\Omega }. \end{aligned}$$Indeed, \(f(x +t \varvec{1}_{\Omega })= f(t(x/t + \varvec{1}_{\Omega }))= tf(x/t + \varvec{1}_{\Omega })=t(f(x/t) + f(\varvec{1}_{\Omega }))=f(x) +tf( \varvec{1}_{\Omega })\).
Under constant additivity, it is easily proved to be equivalent to f submodular on the whole space \({\mathbb {R}}^{\Omega }\).
Indeed \(bv({\mathcal {A}})\) contains the class of all finite games (i.e., \(\Omega \) is finite) since there are finitely many finite chains. Moreover, \(bv({\mathcal {A}})\) contains all capacities v since, using the monotonicity of v, for all finite chains \((A_{k}), \sum \nolimits _{k=1}^{K} |v(A_{k}) - v(A_{k-1}) |= \sum \nolimits _{k=1}^{K} v(A_{k}) - v(A_{k-1}) = v(\Omega ) -v(\emptyset ) = v(\Omega )\). Consequently, \(\Vert v\Vert = v(\Omega )<\infty \). An additive game \(\mu \) is called a charge (or a signed charge) and we point out that the total variation norm of a charge \(\Vert \mu \Vert \) is exactly the variation norm of the game \(\mu \) defined above.
As in Condition (i) of Theorem 2.2.
If f is additive, then the functional f is modular. Indeed, since \(x\vee y+ x \wedge y =x+y\): \(f(x\vee y) +f(x \wedge y) =f(x\vee y+ x \wedge y) =f(x+y)= f(x) +f(y)\) for all x, y in \(B({\mathcal {A}})\).
Define \( \Vert \mu \Vert =|\mu |(\Omega ):= \mu ^{+}(\Omega )+\mu ^{-}(\Omega )\), where \(\mu ^{+}(A):= \sup \{\mu (B) \, :\, B\subseteq A, A \in {\mathcal {A}}\}, \mu ^{-}(\Omega ):= -\inf \{\mu (B) \, :\, B\subseteq A, A \in {\mathcal {A}}\}\) for \(A \in {\mathcal {A}}\). Then \( \Vert \mu \Vert =|\mu |(\Omega )\) (see for example [3] Theorem 2.2.4 page 46, Theorem 4.1.2 page 86, [1] Corollary 10.53 page 397) and the result follows.
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Chateauneuf, A., Cornet, B. Choquet representability of submodular functions. Math. Program. 168, 615–629 (2018). https://doi.org/10.1007/s10107-016-1074-7
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DOI: https://doi.org/10.1007/s10107-016-1074-7