ABSTRACT
The assessment of inequality in resource allocation by means of Lorenz preorders is both well established for univariate distributions and highly problematic for multivariate ones. The main reason for such a state of affairs is the following: if the relevant variables are real-valued, the univariate case allows a natural total ordering of individual endowments, whereas any multivariate distribution, real-valued or otherwise, typically admits only partial rankings (e.g. dominance orderings) of the latter as natural and non-controversial. This problem also arises in a discrete setting, namely when the resources to be allocated amount to a finite set of items/opportunities. That is so because it is by no means obvious if and how the non-controversial set-inclusion partial preorder might be extended to a total preorder of opportunity sets in order to define a Lorenz-like preorder of opportunity distributions amenable to characterizations via simple progressive Pigou-Dalton transfers as established by the classic Hardy-Littlewood-Polya (henceforth HLP) theorem for realvalued (income) distributions. The present chapter is devoted to a critical review of the extant literature on the problem of importing such Lorenz-like preorders in finite settings.
