# 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 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. Sep 25 '15 at 15:30
• @VivekIyer I didn't understand what you meant by "vide Aumann & Shapley". Can you explain? 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. Sep 30 '15 at 15:46

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 & 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

• I am interested in the nonatomic case. What is the reference? Sep 28 '15 at 19:57
• Sorry can't seem to post properly- Try this cruel.org/econthought/essays/edgew/coreaumann.html Sep 29 '15 at 21:09
• Actually, given the fact that one small country has made such a big contribution, maybe we should abandon 'courtesy' for 'sabra rules'! Post the problem- let everyone attack. Sep 30 '15 at 15:55
• Give a generalization of your problem and let sabra vs softies commence! You do understand that nonatomicity is meaningless because in a different complexity class. Sep 30 '15 at 17:39
• Is that your own blog you are promoting in your answer, without disclosure? Oct 1 '15 at 13:39