Rank Robustness of Composite Indices: Dominance and Ambiguity

Many structures in economics - from development indices to expected utility - take the form of a linear composite index, which aggregates linearly across multiple dimensions using a vector of weights. Judgments rendered by composite indices are often given great importance, yet by definition are contingent on an initial vector of weights. A comparison made with one weighting vector could be robust to variations in the weights or, alternatively, it may be reversed at some other plausible vector. This paper presents criteria to discern between these two cases. A general robustness quasi-ordering is defined that requires dominance or unanimous comparisons for a set of weighting vectors, and methods from the Bewley model of Knightian uncertainty are invoked to characterize it. We focus on a particular set of weighting vectors suggested by the epsilon-contamination model of ambiguity, which allows the degree of confidence in the initial weighting vector to play a role. We provide a practical vector-valued representation of the resulting epsilon-robustness quasi-ordering and propose a numerical measure to gauge the robustness of a given comparison. An empirical illustration reports on the robustness of Human Development Index country rankings. We extend our methods to certain nonlinear composite indices and explore the links with decision theory, partial comparability in social choice, and the measurement of the freedom of choice.