On the interpretation of the interaction index between criteria in a Choquet integral model

Abstract

Using a Choquet integral model in the context of multiple criteria decision making often involves assessing a capacity on the basis of preferences among alternatives given by a decision-maker. In such circumstances, the elicited capacity is rarely unique. This lack of uniqueness complicates the interpretation of classic indices, such as the interaction index between criteria. It is often the case that the elicitation makes only use of binary alternatives, i.e., alternative that have either a neutral or a satisfactory evaluation on each criterion. We give conditions guaranteeing that preferences expressed on such alternatives can be represented by a Choquet integral model. On the basis of these conditions, we show that a negative interaction among a group is never necessary, i.e., we can always find a capacity for which this interaction is positive. Outside the framework of binary alternatives, we propose a linear programming model allowing one to test whether the sign of the interaction index remains unchanged for all capacities that are compatible with the preferences expressed by the decision-maker.