A proportional division is a kind of fair division in which a resource is divided among $n$ partners with subjective valuations, and each partner receives a share which is worth for him at least $1/n$ of the total resource value.
This definition is cardinal in nature: it relies on the assumption that each partner has a numeric value function which is unique up to scaling.
Suppose that all we know about the partners is that they have an ordinal preference relation. Is there a natural way to define the notion of proportional fairness in this case?
I thought of several possibilities myself, but I would like to know if something like this has already been done in the literature.