<p>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</p>

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

CC BY-SA 3.0
2015-09-28T19:57:02.920Z
I am interested in the nonatomic case. What is the reference?
Erel Segal-Halevi
CC BY-SA 3.0
2015-09-29T21:09:21.687Z
Sorry can't seem to post properly- Try this http://cruel.org/econthought/essays/edgew/coreaumann.html
Vivek Iyer
CC BY-SA 3.0
2015-09-30T15:55:58.543Z
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.
Vivek Iyer
CC BY-SA 3.0
2015-09-30T17:39:24.733Z
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.
Vivek Iyer
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2015-10-01T13:39:06.023Z
Is that your own blog you are promoting in your answer, without disclosure?
410 gone
CC BY-SA 3.0
2015-10-03T18:36:16.750Z
I have linked to a blog titled 'Poetry as Socioproctology'. I fear your scruples are not so much otiose as illiterate. Economics is supervenient, not reducible to, Maths and Mathis knows that.
Vivek Iyer
CC BY-SA 3.0
2015-10-10T18:40:33.543Z
O.P in a private communication explains he is a Computer Science, not an Economics maven. In other words he is looking for algorithmic, efficiently computable, 'pure strategy' solutions which- from Hamann- Aumann-Shapley et al are known- for Substantivist Econ as a Research Program- to be doomed.
Vivek Iyer
CC BY-SA 3.0
2015-10-10T18:46:45.343Z
The short answer to O.P is- if 'goldlilocks' (or Chichilnesky-Heard) preferences obtain- yes! We can always imagine a condition of 'common knowledge' or 'agency' such that contested notions of rational decision making (most notably Utility max and Regret min; not to speak of Envy freeness, Baumol Superfairness, and Euvoluntary Mike Mungered silliness) are univocal.
Vivek Iyer
CC BY-SA 3.0
2015-10-10T18:48:03.227Z
However, we know from  Alan Kirman and John Maynard Smith that this is a pipe dream. There cab be no 'fair' cake cutting algortihm for Econ iff human evolved by natural selection.
Vivek Iyer
CC BY-SA 3.0
2015-10-10T18:55:34.463Z
Amartya Sen in his own silly way underlines the point. But, the maths of ordinality on posets is not silly. Topological frustration can never be excluded from even the most rigorously- i.e stupidly- specified rubric. Indeed, several currently salient problems can be best modelled thus.
Vivek Iyer
CC BY-SA 3.0
2015-09-25T08:38:34.173Z
2015-09-27T17:34:27.737Z
2015-09-27T17:34:27.737Z
Vivek Iyer
<p>A <a href="https://en.wikipedia.org/wiki/Proportional_division" rel="nofollow">proportional division</a> 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.</p>

<p>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.</p>

<p>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?</p>

<p>I thought of several possibilities myself, but I would like to know if something like this has already been done in the literature.</p>

CC BY-SA 3.0
2015-09-22T18:58:25.403Z
Do you know the following paper? http://link.springer.com/chapter/10.1007%2F978-3-642-30347-0_30
Oliv
CC BY-SA 3.0
2015-09-24T09:03:26.247Z
@Oliv I didn't know about it. I will read. Thanks
Erel Segal-Halevi
CC BY-SA 3.0
2015-09-25T15:30:31.627Z
Do you know the literature on sharing rules for bankruptcy  problems (e.g. http://www.sciencedirect.com.proxy.library.vanderbilt.edu/science/article/pii/S0165489602000707 for a review)? Once there are multiple types of goods to be shared, the cardinality assumption on preferences is often dropped, as in http://www.econ.hit-u.ac.jp/~cces/equlity_and_welfare_2012paper/biung_ghi_ju.pdf. You might find interesting stuff there.
Martin Van der Linden
CC BY-SA 3.0
2015-09-28T19:57:47.883Z
@VivekIyer I didn't understand what you meant by "vide Aumann & Shapley". Can you explain?
Erel Segal-Halevi
CC BY-SA 3.0
2015-09-30T15:46:40.293Z
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
CC BY-SA 3.0
2015-09-30T19:59:31.477Z
O.P does not seem interested in any other answer than the Talmudic Aumann-Maschler equal diviosn of contested element.
Presumably the point is Religious.
Vivek Iyer
CC BY-SA 3.0
2015-10-02T06:13:28.737Z
@VivekIyer In fact, my target application is the problem of fair cake-cutting: https://en.wikipedia.org/wiki/Fair_cake-cutting  in which non-atomicity is usually assumed.
Erel Segal-Halevi
CC BY-SA 3.0
2015-10-03T18:24:37.977Z
Why not mention that? Either your problem was trivial or it comes a cropper on the bugbear of all non-atomicity results- viz. silly things being implied by reliance on Zorn's Lemma.However, there is no canonical Muth rational cake division because regret minimization may be superior to utility maximization as may many other non-deterministic solutions.
Vivek Iyer
CC BY-SA 3.0
2015-10-03T22:48:21.477Z
I see you have misunderstood the Lyapunov candidate function despite the link given by me.
Vivek Iyer
<reference-request><social-welfare><fairness>
Ordinal axiomatization of proportional division
Erel Segal-Halevi
# Ordinal axiomatization of proportional division

(Asked by: username_0 on 2015-09-22)

A [proportional division](https://en.wikipedia.org/wiki/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.

**Tags:** reference-request, social-welfare, fairness

> **Comments:**
> **username_1**: Do you know the following paper? http://link.springer.com/chapter/10.1007%2F978-3-642-30347-0_30  
> **username_0**: @username_1 I didn't know about it. I will read. Thanks  
> **username_2**: Do you know the literature on sharing rules for bankruptcy  problems (e.g. http://www.sciencedirect.com.proxy.library.vanderbilt.edu/science/article/pii/S0165489602000707 for a review)? Once there are multiple types of goods to be shared, the cardinality assumption on preferences is often dropped, as in http://www.econ.hit-u.ac.jp/~cces/equlity_and_welfare_2012paper/biung_ghi_ju.pdf. You might find interesting stuff there.  
> **username_0**: @username_3 I didn't understand what you meant by "vide Aumann & Shapley". Can you explain?  
> **username_3**: 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.  
> **username_3**: O.P does not seem interested in any other answer than the Talmudic Aumann-Maschler equal diviosn of contested element. Presumably the point is Religious.  
> **username_0**: @username_3 In fact, my target application is the problem of fair cake-cutting: https://en.wikipedia.org/wiki/Fair_cake-cutting  in which non-atomicity is usually assumed.  
> **username_3**: Why not mention that? Either your problem was trivial or it comes a cropper on the bugbear of all non-atomicity results- viz. silly things being implied by reliance on Zorn's Lemma.However, there is no canonical Muth rational cake division because regret minimization may be superior to utility maximization as may many other non-deterministic solutions.  
> **username_3**: I see you have misunderstood the Lyapunov candidate function despite the link given by me.  

---

## 1 answers

### Answer by username_3 on 2015-09-25. Score: -2

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>

> **Comments:**
> **username_0**: I am interested in the nonatomic case. What is the reference?  
> **username_3**: Sorry can't seem to post properly- Try this http://cruel.org/econthought/essays/edgew/coreaumann.html  
> **username_3**: 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.  
> **username_3**: 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.  
> **username_4**: Is that your own blog you are promoting in your answer, without disclosure?  
> **username_3**: I have linked to a blog titled 'Poetry as Socioproctology'. I fear your scruples are not so much otiose as illiterate. Economics is supervenient, not reducible to, Maths and Mathis knows that.  
> **username_3**: O.P in a private communication explains he is a Computer Science, not an Economics maven. In other words he is looking for algorithmic, efficiently computable, 'pure strategy' solutions which- from Hamann- Aumann-Shapley et al are known- for Substantivist Econ as a Research Program- to be doomed.  
> **username_3**: The short answer to O.P is- if 'goldlilocks' (or Chichilnesky-Heard) preferences obtain- yes! We can always imagine a condition of 'common knowledge' or 'agency' such that contested notions of rational decision making (most notably Utility max and Regret min; not to speak of Envy freeness, Baumol Superfairness, and Euvoluntary Mike Mungered silliness) are univocal.  
> **username_3**: However, we know from  Alan Kirman and John Maynard Smith that this is a pipe dream. There cab be no 'fair' cake cutting algortihm for Econ iff human evolved by natural selection.  
> **username_3**: Amartya Sen in his own silly way underlines the point. But, the maths of ordinality on posets is not silly. Topological frustration can never be excluded from even the most rigorously- i.e stupidly- specified rubric. Indeed, several currently salient problems can be best modelled thus.  

---

