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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.

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    $\begingroup$ Do you know the following paper? link.springer.com/chapter/10.1007%2F978-3-642-30347-0_30 $\endgroup$
    – Oliv
    Commented Sep 22, 2015 at 18:58
  • $\begingroup$ @Oliv I didn't know about it. I will read. Thanks $\endgroup$
    – Erel Segal-Halevi
    Commented Sep 24, 2015 at 9:03
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    $\begingroup$ 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. $\endgroup$
    – Martin Van der Linden
    Commented Sep 25, 2015 at 15:30
  • $\begingroup$ @VivekIyer I didn't understand what you meant by "vide Aumann & Shapley". Can you explain? $\endgroup$
    – Erel Segal-Halevi
    Commented Sep 28, 2015 at 19:57
  • $\begingroup$ 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. $\endgroup$
    – Vivek Iyer
    Commented Sep 30, 2015 at 15:46

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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

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  • $\begingroup$ I am interested in the nonatomic case. What is the reference? $\endgroup$
    – Erel Segal-Halevi
    Commented Sep 28, 2015 at 19:57
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    $\begingroup$ Sorry can't seem to post properly- Try this cruel.org/econthought/essays/edgew/coreaumann.html $\endgroup$
    – Vivek Iyer
    Commented Sep 29, 2015 at 21:09
  • $\begingroup$ 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. $\endgroup$
    – Vivek Iyer
    Commented Sep 30, 2015 at 15:55
  • $\begingroup$ 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. $\endgroup$
    – Vivek Iyer
    Commented Sep 30, 2015 at 17:39
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    $\begingroup$ Is that your own blog you are promoting in your answer, without disclosure? $\endgroup$
    – 410 gone
    Commented Oct 1, 2015 at 13:39

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