# Axiomatization of an importance index for $k$-ary games

## TL;DR

This paper introduces a new importance index for multi-criteria decision models over discrete attributes, generalizing the concept to $k$-ary games and providing an axiomatic foundation for it.

## Contribution

It proposes a novel importance index for non-monotonically increasing models, extending the theory of $k$-ary games with an axiomatic characterization.

## Key findings

- Classical solutions like the Shapley value are unsuitable for these models.
- The proposed index is based on average variation along attributes.
- An axiomatic characterization of the importance index is provided.

## Abstract

We consider MultiCriteria Decision Analysis models which are defined over discrete attributes, taking a finite number of values. We do not assume that the model is monotonically increasing with respect to the attributes values. Our aim is to define an importance index for such general models, considering that they are equivalent to $k$-ary games (multichoice games). We show that classical solutions like the Shapley value are not suitable for such models, essentially because of the efficiency axiom which does not make sense in this context. We propose an importance index which is a kind of average variation of the model along the attributes. We give an axiomatic characterization of it.