The choice of specific weights for the variables included in composite indices is a most difficult topic that might involve deep ethical considerations. In face of such a daunting task, a group of decision-makers might be uncertain and prefer to allow for a certain degree of weights underspecification. However, allowing for larger sets of admissible weights might lead to increasingly different admissible rankings. In this paper we introduce an axiomatically characterized ranking distance function that is used to explore the pace at which the dissimilarity between different admissible rankings increases as the set of admissible weights that the decision-makers are willing to accept becomes gradually large. This can be very useful in many areas of the social sciences to assess the reliability and robustness of any ranking derived from the values of composite indices when the choice of a specific weighting scheme is controversial.
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