Annals of Operations Research

, Volume 271, Issue 2, pp 701–735

# An axiomatization of the Choquet integral in the context of multiple criteria decision making without any commensurability assumption

• Christophe Labreuche
Original Research

## Abstract

An axiomatization of the Choquet integral is proposed in the context of multiple criteria decision making without any commensurability assumption. The most essential axiom—named Commensurability Through Interaction—states that the importance of an attribute i takes only one or two values when a second attribute k varies. When the importance takes two values, the point of discontinuity is exactly the value on the attribute k that is commensurate to the fixed value on attribute i. If the weight of criterion i does not depend on criterion k, for any value of the other criteria than i and k, then criteria i and k are independent. Applying this construction to any pair ik of criteria, one obtains a partition of the set of criteria. In each block, the criteria interact one with another, and it is thus possible to construct vectors of values on the attributes that are commensurate. There is complete independence between the criteria of any two blocks in this partition. Hence one cannot ensure commensurability between two blocks in the partition. But this is not a problem since the Choquet integral is additive between subsets of criteria that are independent.

## Keywords

Choquet integral Capacity Commensurability Interaction between criteria

## Notes

### Acknowledgements

The author would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.

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