The Choquet integral for the aggregation of interval scales in multicriteria decision making

This paper addresses the question of which models fit with information concerning the preferences of the decision maker over each attribute, and his preferences about aggregation of criteria (interacting criteria). We show that the conditions induced by these information plus some intuitive conditions lead to a unique possible aggregation operator: the Choquet integral.

💡 Research Summary

The paper investigates how to construct a decision‑making model that faithfully reflects a decision maker’s (DM’s) preferences when multiple criteria are involved. The authors start from the classical two‑step utility representation u(x)=F(u₁(x₁),…,uₙ(xₙ)), where each uᵢ maps a single attribute Xᵢ onto a common satisfaction scale and F aggregates these scores. Traditional approaches, notably the MacBeth methodology, assume that each uᵢ is a difference (interval) scale and that the aggregation function is a weighted sum. While this works for additive preferences, it cannot capture interactions among criteria.

To overcome this limitation, the authors propose to collect two kinds of information from the DM:

1. Intra‑criterion information – For each criterion i, the DM identifies a worst element 0ᵢ and a best element 1ᵢ, which serve as absolute reference points. The DM is also asked to assess the intensity of preference between any two acts that differ only on criterion i (i.e., (xᵢ,0_{‑i}) vs. (yᵢ,0_{‑i})). These assessments define a difference scale uᵢ satisfying:

   • Order preservation (if (xᵢ,0_{‑i}) is preferred to (yᵢ,0_{‑i}) then uᵢ(xᵢ) ≥ uᵢ(yᵢ)).
   • Proportionality of differences (the ratio of two preference differences is constant across comparable pairs).
   • Normalization uᵢ(0ᵢ)=0, uᵢ(1ᵢ)=1.
   • Consistency of proportionality (transitivity of the ratio constant).

2. Inter‑criterion information – The DM evaluates the overall satisfaction of “all‑or‑nothing” bundles (1_A,0_{‑A}) for every subset A⊆N, where 1_A denotes that criteria in A are at their best and the rest are at their worst. From these judgments a set function µ: 2ᴺ→