A Discrete Choquet Integral for Ordered Systems

TL;DR

This paper introduces a generalized discrete Choquet integral for arbitrary finite set systems, extending classical models and providing computational algorithms and supermodularity characterizations for specific system classes.

Contribution

It develops a new model for the Choquet integral applicable to ordered set systems, including algorithms and supermodularity characterizations.

Findings

The model includes classical Choquet integral as a special case.

A simple Monge-type algorithm computes the integral for intersection systems.

Superadditivity is characterized via supermodularity on union-closed systems.

Abstract

A model for a Choquet integral for arbitrary finite set systems is presented. The model includes in particular the classical model on the system of all subsets of a finite set. The general model associates canonical non-negative and positively homogeneous superadditive functionals with generalized belief functions relative to an ordered system, which are then extended to arbitrary valuations on the set system. It is shown that the general Choquet integral can be computed by a simple Monge-type algorithm for so-called intersection systems, which include as a special case weakly union-closed families. Generalizing Lov\'asz' classical characterization, we give a characterization of the superadditivity of the Choquet integral relative to a capacity on a union-closed system in terms of an appropriate model of supermodularity of such capacities.