Ranking by weighted sum

Abstract

When choosing an alternative that has multiple attributes, it is common to form a weighted sum ranking. In this paper, we provide an axiomatic analysis of the weighted sum criterion using a general choice framework. We show that a preference order has a weak weighted sum representation if it satisfies three basic axioms: Monotonicity, Translation Invariance, and Substitutability. Further, these three axioms yield a strong weighted sum representation when the preference order satisfies a mild condition, which we call Partial Representability. A novel form of non-representable preference order shows that partial representability cannot be dispensed in establishing our strong representation result. We consider several related conditions each of which imply a partial representation, and therefore a strong weighted sum representation when combined with the three axioms. Unlike many available characterizations of weighted sums, our results directly construct a unique vector of weights from the preference order, which makes them useful for economic applications.