I have been interested in fairness recently, both fairness in classification and the more traditional setting of fairness in resource allocation. Defining fairness is a tricky thing. Consider for example a resource to be divided between three people with equal claims for ownership. What is the fair division? It sounds natural that equal distribution of the resource (each person getting one third of the resource) is the fairest division. And if one person’s claim of ownership is twice that of each one of the other two, we may consider proportional distribution as the fairest (suggesting that the shares should be half, quarter and quarter). But just like privacy, fairness is a matter of social values (put differently, a matter of ideology). In the above example, we may instead adopt the approach that “From each according to his ability, to each according to his need” [Karl Marx, Critique of the Gotha Program, 1875]. Namely, perhaps the fair division is not proportional to the claims of ownership but rather proportional to the justifiable demand. Today, I’d like to discuss a different and intriguing approach (pointed out to me by Moni Naor), which precedes Karl Marx as well as Milton Friedman. Today we are going all the way back to the Mishnah and Talmud.
As my knowledge of the Mishnah and the Talmud is close to nil, my post is based on other sources (but the errors are all mine): (1) Gadi Aleksandrowicz’ blog which is in Hebrew. if you can read Hebrew its recommended you continue the reading there (while you are in his blog, there is much more to read). Gadi was also gracious enough to allow me use his pictures. (2) Directly from one of the two protagonists in our story, we have Aumann’s manuscript in English. So the only reason to continue reading here is if you are too lazy to click … 😉
Consider a man who dies and has three creditors (originally in the Mishnah, his three bereaved wives). How should his estate be divided if it is smaller than the claims of the creditors? The Mishnah deals with this very situation and specifies the right division in three examples which are summarized by the following table:
| Creditors’ Claims: | 100 | 200 | 300 |
| Estate’s Size: | |||
| 100 | 33 1/3 | 33 1/3 | 33 1/3 |
| 200 | 50 | 75 | 75 |
| 300 | 50 | 100 | 150 |
The Mishnah represents Rabbinic Law (law that is not explicitly specified in The Bible but rather follows the interpretation of The Bible by the Rabbinic Establishment). But what does this law mean? What is the rule that is followed here? How can we generalize these examples to different sums or different number of creditors? The first row seems to indicate equal division of the estate (independent of the claim of each creditor), the third row seem to indicate division which is proportional to the claims, and it’s hard to figure out what is the rule followed by the second row.
Indeed, generations of interpreters (over a period of almost 2000 years) did not give a convincing explanation of the rule governing this Mishnah. Recently though, Aumann and Mashler (Journal of Economic Theory 36 (1985), pp. 195-213) figured it out! While the way Aumann and Mashler reached their explanation was based on game theoretic concepts, they were later able to explain the Mishnah in a way that is so natural that its almost hard to understand how it eluded the interpreters for so long (of course, it is important to realize that a solution that is elegant and simple in retrospect is many times very hard to come by).
To understand the rule applied by the Mishnah we need to look at the way the Mishnah suggests splitting a disputed asset (such as an estate) between two contestants. A famous Mishnah (which even I knew about) says:
“Two hold a garment; … one claims it all, the other claims half. … Then the one is awarded ¾, the other ¼”.
So if there is a dispute over $100 and one creditor claims the entire $100 while the other claims $50 the first will get $75 and the second $25. Just like in the previous case, the rule behind this example is not that intuitive. Fortunately, the interpreters of the Mishnah were able to explain the rule well. Consider the second creditor: he only has a claim for $50, so the $50 that he has no claim over can be safely given to the first creditor. We are left with $50 to divide and these $50 are contested (each creditor has a claim over them). This $50 should therefore be equally split giving an overall split of $75 / $25. Quoting Aumann:
“The principle here is equal division of the contested sum.”
Note that this division is not necessarily more just than other divisions (such as the proportional division), but it seems clear that this is what the Mishnah had in mind (and Aumann demonstrates other places in the Mishnah where this principle governs). Back to our example with three creditors, it can be seen that if we focus on any two creditors and analyze the division (of the remaining sum after deducting the third share) we have that it follows the principle of “equal division of the contested sum:”
| Creditors’ Claims: | 100 | 200 |
| Estate’s Size: | ||
| 66 2/3 | 33 1/3 | 33 1/3 |
| 125 | 50 | 75 |
| 150 | 50 | 100 |
| Creditors’ Claims: | 100 | 300 |
| Estate’s Size: | ||
| 66 2/3 | 33 1/3 | 33 1/3 |
| 125 | 50 | 75 |
| 200 | 50 | 150 |
| Creditors’ Claims: | 200 | 300 |
| Estate’s Size: | ||
| 66 2/3 | 33 1/3 | 33 1/3 |
| 150 | 75 | 75 |
| 250 | 100 | 150 |
Aumann and Mashler give the following rule for dividing an estate to arbitrary number of creditors: the rule is simply that the division should observe the principle of “equal division of the contested sum” as far as any two creditors are concerned. Is there always such a division? Yes! Could there be more than one such division? No! Is there an efficient way to come up with such a division? Yes! Furthermore, any other solution would be inconsistent with the principles of the Mishnah in a closely related context and therefore it seems fair to say that Aumann and Mashler got it absolutely right.
There is a clever way to demonstrate that the Aumann and Mashler uniquely defines a division through the Rule of Linked Vessels. Let me conclude with a few hints and you are welcomed to read more in the two sources I previously recommended: Hebrew, English.
Let’s start with two creditors, and consider the following linked tubes:
The height of the two tubes is identical and so is their width (and they are connected in the base by a very narrow tube of negligible volume). Each tube is composed of two identical halves connected with a very narrow tube (of negligible volume). The volume of each tube corresponds to the claim of the corresponding creditor. Finally, we pour liquid into the linked tubes of volume which corresponds to the size of the estate. As the liquid is divided to the two tubes, this corresponds to a division of the estate to the two creditors. Aumann argues via a rather simple case analysis that this division is the unique one that satisfies the principle of “equal division of the contested sum.”
Now consider any number of creditors and the corresponding linked tubes. For example, for four creditors we will have something like that:
It’s obvious that the division we get satisfies the principle of “equal division of the contested sum” for every two creditors (as the liquid in every pair of linked tubes reaches identical height). Furthermore, it is also clear that it is the unique such division as for every other division there will be a pair of tubes where the liquid reaches different heights. Nice!
Windows On Theory 
I understand the rule here, but in what sense is it a good one?
Good question, and in what sense is proportional division a good (or just) rule? (Dealing with Jewish sources, its only appropriate to answer a question with a question 😉 )
For any rule there can be a debate if it is fair or not, and if it makes sense or not. But one thing is clear: it is desirable that the rule results in a unique prediction, and indeed in this case we have both existence and uniqueness as one would hope.
I still cannot answer about the rational of this method (Why we may consider it just? What kind of social behavior it encourages?), and I wish there was more discussion of this aspect available (is there?). Nevertheless, there are some properties of this method which I like to point out. The example I give above (with the disputed $100) gives the feeling that the principle of equal division of the contested sum favors the “large creditors”. Indeed, the creditor with a claim of $50 gets half of his claim rather than two thirds. But this is in some sense the worse case for small creditors. What I mean is that the principle actually favors small creditors until half of their claim is satisfied (and then they get lower priority). The linked tubes characterization implies that if a creditor gets x which is less than half its claim than no other creditor gets more than x.
Consider now for example an estate of 1 million dollars with a total debt of 10 million dollars. Assume that 9 million dollars are owed to the bank and 1 million is owed to small creditors. If we use proportional division, a creditor who invested his life savings of 100k$ will get 10k$ back. The bank will lose 8.1 million of its money. (In reality the small creditors may get nothing.) Using the principle described in this post the small creditor will get 50k$ back and the bank will lose 8.5 million. Somehow, there is something more pleasing to me in this second scenario (but granted, I did devise this example for exactly this effect 🙂 )