Choquet integral in decision analysis - lessons from the axiomatization

Abstract

The Choquet integral is a powerful aggregation operator which lists many well-known models as its special cases. We look at these special cases and provide their axiomatic analysis. In cases where an axiomatization has been previously given in the literature, we connect the existing results with the framework that we have developed. Next we turn to the question of learning, which is especially important for the practical applications of the model. So far, learning of the Choquet integral has been mostly confined to the learning of the capacity. Such an approach requires making a powerful assumption that all dimensions (e.g. criteria) are evaluated on the same scale, which is rarely justified in practice. Too often categorical data is given arbitrary numerical labels (e.g. AHP), and numerical data is considered cardinally and ordinally commensurate, sometimes after a simple normalization. Such approaches clearly lack scientific rigour, and yet they are commonly seen in all kinds of applications. We discuss the pros and cons of making such an assumption and look at the consequences which axiomatization uniqueness results have for the learning problems. Finally, we review some of the applications of the Choquet integral in decision analysis. Apart from MCDA, which is the main area of interest for our results, we also discuss how the model can be interpreted in the social choice context. We look in detail at the state-dependent utility, and show how comonotonicity, central to the previous axiomatizations, actually implies state-independency in the Choquet integral model. We also discuss the conditions required to have a meaningful state-dependent utility representation and show the novelty of our results compared to the previous methods of building state-dependent models.