# 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.

• Do you know the following paper? link.springer.com/chapter/10.1007%2F978-3-642-30347-0_30 – Oliv Sep 22 '15 at 18:58
• @Oliv I didn't know about it. I will read. Thanks – Erel Segal-Halevi Sep 24 '15 at 9:03
• Do you know the literature on sharing rules for bankruptcy problems (e.g. sciencedirect.com.proxy.library.vanderbilt.edu/science/article/… for a review)? Once there are multiple types of goods to be shared, the cardinality assumption on preferences is often dropped, as in econ.hit-u.ac.jp/~cces/equlity_and_welfare_2012paper/…. You might find interesting stuff there. – Martin Van der Linden Sep 25 '15 at 15:30
• @VivekIyer I didn't understand what you meant by "vide Aumann & Shapley". Can you explain? – Erel Segal-Halevi Sep 28 '15 at 19:57
• Robert Aumann and Harlow Shapley are Nobel Prize winning Economists. Aumann-Shapley values are germane in division problems. However this is not a 'natural' solution for reasons disclosed by the theory. Kindly have the courtesy to up vote before requesting more info- especially if your knowledge base is non conventional. – Vivek Iyer Sep 30 '15 at 15:46