Ordinal axiomatization of proportional division 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.
The answer hinges on the level of 'atomicity' dictated by the problem. If the decision making unit is non atomic- i.e. can make infinitesmal decisions- then there is always an algorithm corresponding to the competitive allocation such that it is known also to be a 'cardinal' value allocation under certain conditions- vide Aumann &amp; Shapley. However this would not be robust to even a small perturbation towards atomicity. My understanding is that this field is interesting because of how small changes in the lower limit to what is within the agent's decision making scope can lead to big changes in behavior

D.G Saari's work may be of interest if only as an awful warning- vide http://socioproctology.blogspot.co.uk/search?q=dark+matter