Welfare and Inequality Comparisons for Uni- and Multi-dimensional Distributions of Ordinal Data

Decision makers and social planners are often faced with a problem of evaluating distributions of ordinal variables i.e. variables for which there are no numbers but only the ordering, such as, for example, self-reported health status, life satisfaction, working environment, quality of public goods, living conditions etc. Standard tools, namely, stochastic dominance, and inequality and risk measures, produce conclusions that can be reversed depending on the cardinalisation of an ordinal indicator which is arbitrary. Utilising the notion of integration on partially ordered sets we extend the well-known Hardy et al. (1934) result to an ordinal setting, both univariate and multivariate.