Social welfare with incomplete ordinal interpersonal comparisons

Let X be a set of “personal states”; any person, in any circumstance, is at some point in X. A social state assigns an element of X to every person in society. Suppose it is sometimes possible to make ordinal interpersonal comparisons of well-being. We represent this with a (possibly incomplete) preorder on X. From this, we can derive a (possibly incomplete) preorder on the set of social states, which ranks them in terms of their aggregate welfare. We define the appropriate analogs of the maximin and leximin social welfare orders in this framework, and axiomatically characterize them.