Social well-being is intrinsically multidimensional. Welfare indices attempting to reduce this complexity to a unique scalar measure abound in many areas of economics and public policy. Ranking alternatives based on such measures depends, sometimes critically, on how the different dimensions of welfare are weighted. In this paper, a theoretical framework is presented that yields a set of consensus rankings in the presence of such weight imprecision.

The main idea is to consider a vector of weights as an imaginary voter submitting preferences over alternatives. With this voting construct in mind, the well-known Kemeny rule from social choice is introduced as a means of aggregating the preferences of many plausible choices of weights, suitably weighted by the importance attached to them. The axiomatic characterization of Kemeny’s rule is shown to extend to the present context, and a graph-theoretic algorithm is developed to efficiently compute or approximate the set of Kemeny-optimal rankings. An analytic solution is derived for an interesting special case of the model corresponding to generalized weighted means and the epsilon-contamination framework of Bayesian statistics. The model is applied to the ARWU index of Shanghai University, a popular composite index measuring academic performance.