How Game Theory Solved a Religious Mystery
[Update: This article has been edited for clarity due to its surprising popularity.]
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The Bankruptcy Problem
A man owes debts of 100, 200, and 300, but dies with insufficient funds to pay everyone.
How should his estate be divided?
As we all know, there might not be one correct answer. Fair division is a concept that depends as much on logic as it does on social custom. To see why, consider the following three situations that afford very different solutions:
- A parent promises gifts to his children, but has to back off when a bonus is smaller than expected
- A publicly traded company issues stock and bonds, but soon goes bankrupt in an accounting scandal
- Partygoers order items at a restaurant, with promises to pay, and then end up arguing over the best way to split the bill
There isn’t a single right way to approach any of these problems. That’s what family fights, lawsuits, and restaurant arguments demonstrate every day. The conflict is a matter of perspective.
Some people prefer proportional division that depends on debt size. An example is the classic “pay what you ordered” method in restaurants where guests put money based on food they ordered. As logical as this sounds, not everyone desires this method.
Others prefer splitting things up equally. They argue it is the person–not the debt size–that matters. Equal division is common particularly among families with young children. During Christmas or holiday time, parents may choose to give every child the same gift regardless of age or merit.
What gets accepted depends on social custom. Getting everyone to agree is an exercise in persuasion, not in economics. It’s possible for emotionally pleasing methods to beat more sound systems.
One of the earliest discussions of fair division comes from the Babylonian Talmud, a record of discussions about Jewish laws and customs. The Talmud contains discusses a bankruptcy problem in the context of a man offering debts to his wives in excess of his assets. The Talmud answer is not immediately obvious, and in fact, the answer baffled academics for over almost 2,000 years. Let’s see why.
The Talmud answer
How should an estate be divided among three creditors claiming sums of 100, 200, and 300?
The Talmud offers answers through three examples. The text does not contain a general rule, which is what makes these answers seemingly contradictory. The three cases are when the estate size is 100, 200, and 300.
In the first case when the estate size is 100, the Talmud awards 33 1/3 to each party. The division suggests a principle of an equal division, which is easy mathematically and holds social appeal. But strangely this is not the same idea used in the other cases.
In the third case of 300, the Talmud offers a division of 50, 100, and 150. The math here is a proportional division based on the size of the debt. In modern times, proportional division holds wide appeal among lawyers and economics. At this point one might ask why is the 300 case treated differently than the 100 case?
If that question bothers you, then get ready for another surprise in the division for 200. In this case, the estate is supposed to be divided as 50, 75, and 75. Not only does the division not classify as an equal division nor a proportional division, but it is simply a curious decision altogether. Why should the second and third creditors be given the same amount of money? And where do the numbers come from?
Before I proceed, it’s worth summarizing the claims in a table. We can think about the Talmud answers as a table that illustrates how an estate would be divided. I provide an illustration below, in which the rows are estate sizes, the columns are claims, and the table entries are the division size.
The answers defied a proper explanation for almost 2,000 years, filling volumes of critical review. Some scholars have essentially given up and suggested the 200 case might be an issue of faulty transcription. And this is the unlikely background for which game theory enters and possibly saves the day.
Game theory offers an answer
In the 1980s, Professors Robert Aumann and Michael Maschler wrote a paper claiming to have cracked the mystery.
They suggest there is no inconsistency in the Talmud answer. Aumann and Maschler demonstrate the Talmud answer can be viewed as a consistent application of a game theory principle. Why was game theory used? It turns out the Talmud answer is the solution (the nucleolus) of a properly defined coalitional game. Aumann and Maschler explain the concept in lay terms as a single and consistent principle: equal division of the contested sum.
It is worth being skeptical before proceeding. Is the explanation simply a coincidence? After all, there are probably an infinite number of explanations that might produce the same split.
Aumann and Maschler justify their answer by examining other Talmudic passages and suggesting the same principle is applied in many topics. Equal division of the contested sum was apparently a social custom and that would help explain why it might seem strange to us but could have been natural for their culture.
Below I explain the concept of “equal division of the contested sum,” and describe why the Talmud answer demonstrates it.
(Notes for the sources: The original academic paper is in Journal of Economic Theory 36 (1985), pp. 195-213. I have not read that. Instead, I relied on a complementary non-technical article “Game Theory in the Talmud,” written by Robert Aumann, retrieved from Professor Jacob Rosenberg’s website. I also wish to credit Paul Walker’s chronology of game theory, the document that introduced me to this fascinating problem).
Equal division of contested sum, two people
The Talmud examines a situation that might have been common to their times. Suppose two people are arguing over a garment. One claims half belongs to him while the other claims the whole is his. A judge is asked to decide who gets what. What would you do?
There are naturally various answers. One could propose an even split (1/2, 1/2) or a proportional split (1/3, 2/3).
But the Talmud offers a different answer, an answer that turns out to be an equal division of the contested sum (1/4, 3/4). How does this principle work? There are three stages. First, decide what portion of the cloth is being disputed. In this case, exactly half of the garment is being claimed by both parties. Second, split the disputed division among both parties–so 1/4 of the cloth is awarded to each. And third, give the remaining cloth–the “undisputed” portion–entirely to the person whose claim is not disputed.
This logic yields a split of 1/4 for the person claiming half of the garment and 3/4 for the person claiming the whole.
This answer might seem strange, but remember that division methods depend on social custom.
The same method can be used for any problem among two parties, using the same three steps above:
- Determine which portion is contested or claimed by both parties
- Split the contested portion equally
- Assign the uncontested portion to the sole person claiming it
How else might this principle be applied? It can actually be applied to many situations, like when the claims are larger than the asset to be divided, as in the case of dividing an estate.
Equal division of contested sum, two creditors
It’s worth going through a few examples to get a feel for the idea. Let’s examine how to divide estates of various sizes with two creditors claiming 100 and 300.
Example 1: (estate 66 2/3)
If the estate is 66 2/3, then the entire estate is contested. The split should be even at 33 1/3 going to each party.
Example 2: (estate 125)
If the estate is 125, then the first 100 is contested by both parties and divided evenly. The remaining 25 is entirely awarded to the 300-claimant. Hence, the division is 50 and 75.
Example 3: (estate 200)
Finally, if the estate is 200, then again the first 100 is contested by both parties and divided evenly. The remaining 100 is entirely awarded to the 300-claimant. Hence, the division is 50 and 150.
Here are the divisions in tabular form:
Why stop there? Here are some examples when the claims are (100, 200) and (200, 300). Note that these are the remaining pairs of claims for the three-person split that is motivating this article.
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Explaining the Talmud puzzle
Let’s go back to the Talmud division for the three creditors. In the case of a 200 estate, the division was 50, 75, and 75 for parties that claimed debts of 100, 200, and 300.
To analyze this answer, let’s do the following exercise. Take any two creditors and consider how they might split the total money awarded to them. Why would we do that? It’s a check of consistency. It makes sense that pairs of creditors should have claims divided in a manner consistent with the way a disputed garment would be divided.
Consider the pair of creditors claiming 100 and 200. Together they are awarded a sum of 125. How is that sum split? It is split as 50 and 75. And amazingly, that’s matches the work we did in examples above: this answer is consistent with an equal division of the contested sum! The logic is that the first 100 is contested by both parties and split evenly, and the uncontested 25 is awarded to the 200-claimant.
In fact, the same observation can be seen when considering other pairs of creditors. Look at how much the 100 and 300 parties are getting. Together they receive a sum of 125, and this is split as 50 and 75. Again, this answer is consistent with an equal division of the contested sum.
Finally, consider the total reward to the 200 and 300 parties. In this case, the total sum of 150 is split as 75 to each. As the total sum is contested, this once again reflects an equal division of the contested sum.
In other words, when the mysterious Talmud solution is broken down by pairs of creditors, there is a consistent principle. I think this is quite remarkable.
Aumann and Maschler demonstrate the method can be extended, whether the claims are for three creditors, a hundred creditors, or even a million creditors. The same condition needs to be met: the assets are divided up such that the amount received by any two people reflects the principle of equal division of the contested sum. Furthermore, the division is a unique solution.
An algorithm
It’s good enough to see certain divisions are pairwise equal divisions of contested sums. But how do you find them starting from scratch?
Aumann and Maschler show there is in fact only one division that is consistent. And this answer can be described by the following seven step algorithm:
- Order the creditors from lowest to highest claims.
- Divide the estate equally among all parties until the lowest creditor receives one half of the claim.
- Divide the estate equally among all parties except the lowest creditor until the next lowest creditor receives one half of the claim.
- Proceed until each creditor has reached one-half of the original claim.
- Now, work in reverse. Start giving the highest-claim money from the estate until the loss, the difference between the claim and the award, equals the loss for the next highest creditor.
- Then divide the estate equally among the highest creditors until the loss of the highest creditors equals the loss of the next highest.
- Continue until all money has been awarded.
Here is how the claims would be divided in the Talmud example:
Mystery solved? I think so. Not only do the Talmud answers follow a consistent principle, but they also rely on an idea that was very likely a social custom.
In that case, it is surely an interesting case that a tool of logic and rationality–game theory–was needed to decode the Talmud solution, which primarily depended on social custom.
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29 Responses to “How Game Theory Solved a Religious Mystery”
Presh writes:
1. Determine which portion is contested or claimed by both parties
2. Split the contested portion equally
3. Assign the uncontested portion to the sole person claiming it
I am not sure that I follow this. If you and are arguing over ownership of a farm, and I say that I own 1/2 of it, while you say you own all of it, then we arguing about the entire farm.
Perhaps the better way to say it is this. If I argue for division x:y and you argue for a:b, then if y
By Michael Webster on Jun 10, 2008
@Michael Webster If contested property cannot be divided, it could still be assigned a dollar amount. Say the farm you and Presh are arguing over is worth $200m. According to the solution above, you would get $50m of it, and Presh would get $150.
That means that in order to get the whole farm, Presh would have to give you $50m or you would have to give Presh $150m. If neither can buy out the other, then you might have to sell the farm, and buy smaller farms that each of you could afford.
This was a great article. I totally see its applications. Unfortunately, getting us out of the current US mortgage crisis doesn’t appear to be one of them. Do you have a game theory solution to solve that?
By Mike on Jun 10, 2008
For some reason, my first comment was cut off and makes no sense.
I didn’t understand the notion of the “uncontested part”
Here is my own analysis.
If I argue for division x:y and you argue for a:b, then if y is less than b, then I agree that you are at least entitled to y. And since you believe that your are entitled to b, bigger than y, then you agree that your are entitled to y.
I suppose that in some sense you getting y is not contested - but it is a bit weird.
By Michael Webster on Jun 10, 2008
@Mike, who writes: “Unfortunately, getting us out of the current US mortgage crisis doesn’t appear to be one of them. Do you have a game theory solution to solve that?”
Sure, turn the principal in subprime mortgages that cannot be renewed into loans to pay for the renting of the house, and maintain fractional ownership.
Automatic switch from housing stock to rental stock, no widescale foreclosures except when the house couldn’t be rented at any price, and give the housing market a chance to recover,
By Michael Webster on Jun 10, 2008
In editing, it appears I left about a juicy detail about the Talmud example. The three creditors fighting over the man’s estate are his three wives. That goes to show you how much social custom plays into the division of the estate.
Michael Webster: I can see why “uncontested” is an imprecise term, but Mike offers the appropriate intuition.
The person claiming half the farm provides an unstated agreement that the other half belongs to the person claiming it all–that’s the “uncontested” part.
The parties are also not offering up divisions, per se, but rather claims. This makes most sense in the Talmud example where the creditor claims were already written in a contract.
Thanks for your suggestion about the housing market.
Mike: Thanks for the clarification to Michael Webster and for the feedback on the article.
Today I came up with an example of how equal division is sort of used–in restaurants. After people pay what they owe, there is sometimes a shortage of funds. Usually this is settled by equal division. In all, the bill gets paid by an equal division of the *unclaimed* portion of the bill.
Equal division does have a major flaw: it leads everyone to inflate their claims. There is a direct increase in rewards to higher claims. It works best when the claims are already written in a contract.
By Presh Talwalkar on Jun 10, 2008
Didn’t know the Talmud had that in it–fascinating. Good explanation too, though a tad long. Luckily, I’m getting paid by the hour here :p
By Sam on Jun 16, 2008
Sam: Glad I can help you expand your work time
I guess the article was long since it’s kind of a strange concept, like calculus. Once you “get it,” however, you wonder why it was so hard to understand in the first place.
By Presh Talwalkar on Jun 17, 2008
The last table (contested_debts_various_estates.png) has a typo when the estate size is 250. It doesn’t add up correctly. The 300 guy should be receiving only 100.
By Eric on Jun 19, 2008
Eric: Thanks for telling me the typo–should be fixed now.
By Presh Talwalkar on Jun 19, 2008
I can’t believe scholars never thought of this.
My parents had three kids, and we kids often found the issue of how to divvy things up to be fascinating. We tried to think up all the ways it could be done, and came up with “equal division of the contested sum,” fairly early, and then tried to consider how to play it out in different situations.
I didn’t hear of game theory until many years later. This is not a counter-intuitive or difficult to understand problem, nor is it hard to invent. I’m not that smart, and scholars can’t be that dumb. I don’t get it.
By Dave L on Aug 21, 2008
Sorry, I don’t follow. The examples have been carefully worked out to produce reasonable results but all we need is to change them a little to show how they don’t work. Suppose the estate size is 140. What happens is that the creditor with the lowest claim receives more (50) than the creditors with higher claims (45 each). It makes no sense.
The way I see it, by applying the principle of division of the contested sum the resoults should be:
For an estate size of 100, 100 contested by all the creditors; it is divided equaly and each one receives 33 1/3.
For an estate size of 200, 100 is contested by all of the creditors; all of them receive 33 1/3 on those 100. Another 100 is only contested by two creditors; each one of these receives an equal share, 50 for each one of them. In this case the lowest claim receives 33 1/3, the two other claimants receive 83 1/3 each.
For an estate size of 300, 100 is contested by the three, each one receives 33 1/3. Another 100 is contested by two, each one of them receives 50 on those 100. The last 100 is contested by only one of the creditors, he receives it entirely. In this case the lowest claimant still receives 33 1/3, the middle claimant receives 83 1/3 and the highest claimant receives 183 1/3.
For me this is what makes sense according to the principle of division of the contested sum. The way you present it is not consistent with the principle or at least that’s what seems to me.
By Sergio on Aug 21, 2008
Dave L: I am getting my history via Aumann who claims he’s the first to figure it out. With religious texts one has to put the work in context of other examples. Not all examples (though most) in the Talmud follow this principle so that is why the answer was elusive.
By Presh Talwalkar on Aug 22, 2008
Sergio: Thank you for a detailed comment. Let me address your questions.
In your example, with an estate of 140, the claims would be divided 46 2/3 to each. The algorithm specifies each person gets an equal amount until the lowest creditor gets 1/2 of his claim. Then the remaining people start getting money and you inductively proceed.
Read Aumann’s original for a better explanation than mine (linked in this article as “Game Theory in the Talmud”…linking in the comment is not working)
By Presh Talwalkar on Aug 22, 2008
I believe the new way of doing this is much simpler. The government takes all of it.
By Ben on Aug 22, 2008
Because this division rule divides the contested amount equally, each player’s best response is always to claim as much as they can legally. This is why, you should not claim half the farm, but all the farm knowing , anticipating that others will claim it all too.
Then, when the property right is not based on a precedent (lets say years of marriage which makes you eligible for more share of an inheritance) decision rule comes to “divide equally”.
In many countries when a person dies his possessions are not distributed based on a will written by the dead men but according to some law. In Turkey for example you cannot leave more than a certain percentage of your possessions to other than family and inheritance is distributed based on some formula. For example wife is eligible for 50 percent of the will if they had children and to all of the will if there are no kids. If there is one kid, they share half half, if there are 2 kids then wife gets 50 percent and kids divide the rest.
I believe the decision rule here can be used in such cases where everybody cannot claim all.
By pangaean on Aug 22, 2008
I formed sentiments very similar to Sergio’s as I read through this article; having read Aumann’s original like you advised him to do I’m still confused. Was the basis for this concept to simply explain the situation presented in the Talmud or is to do that as well as to find a practical division of an estate for creditors? Aumann states that the old system was to divide by the dollar not the person yet it seems that’s exactly what he does as the lowest creditor gets a proportionately larger amount than does the highest creditor. If the estate is 100 for the three debtors claiming 100, 200, and 300 then like the garment example they all claim the whole and therefore the whole should be divided up evenly. But when the estate is 200 why does it not logically follow that the same principle be applied?- 33 1/3 go to each and then the remaining 100, which is now only contested by two parties, divided into two. This process where the 100 gets 50 and the other two get 75 each doesn’t really mesh with the “principle of equal division of the contested sum” because it’s not equal in anyway that I’m understanding. I hope this doesn’t seem to be redundant compared to Sergio’s comment, and thanks in advance for any clarification you could provide.
By Gideon L. on Aug 24, 2008
pangean:
You are right on the money! The decision rule here works when there is a previous written agreement about the claims–the creditors were written in the will, for instance. I was fascinated by the example in Turkey and how this can easily turn into a case of equal division, if claims can be inflated.
Gideon L.:
Thanks for the question–I’ll try to explain. I think Aumann was trying to establish the Talmud division could be seen as the consistent application of one rule–equal division of the contested sum. He goes through historical examples to provide corroborating evidence.
I can understand why it’s hard to get around the solution for the case of three people. What we need to keep in mind is it is more a principle, or a condition, than a rule. Let me explain.
Why does the 50/75/75 case work? The reason is you need to look at any pair of creditors. The 100/200 creditors get 125 jointly, which would be split as 50/75. The 200/300 creditors get 150 jointly, which would be split as 75/75. The 100/300 creditors get 125 jointly, which would again be split as 50/75. It’s easy to verify these pairs.
So the question is then how can you get to 50/75/75 from the start? To do that you use an algorithm–keep adding money equally until someone reaches 1/2 of their claim. Then stop giving money to them and share amongst the remaining parties until the second lowest claim person gets 1/2 of their claim. Continue until each has 1/2 of their claim, and then essentially work backwards.
Hope this clarifies things.
By Presh Talwalkar on Aug 26, 2008
Why do you continue until someone reaches half their claim?
The way I was reading it, before the algorithm, it would have implied that the cases are as follows:
Case 200
100 Collector claims rights to half
200 and 300 Collectors claim rights to entire 200
Therefore, the first 100 would be divided equally, 33 1/3 to each.
The second 100 would be divided between 200 and 300 claimaints, yieldings 66 2/3 to each.
100 gets 33 1/3
200 gets 66 2/6
300 gets 66 2/6
And the 300 case would play out as
100 gets 33 1/3
200 gets 66 2/3
300 gets 166 2/3
I understand the algorithm would yield different answers - what I’m unsure of is why the algorithm applies - it changes the rules from the basic principle.
By Tim Weber on Aug 26, 2008
Tim Weber:
I can see there is still much confusion of this topic, and I appreciate your question and comment. Teaching is my passion, so let me try once again!
I’ll start by explaining why the proposals you offer do not work. Take the case of 200. In your split, the 100/200 claimants receive a total of 100 divided as 33 1/3 and 66 2/3. But if you think about it, all 100 is contested between them. This means each should receive 50/50, if the total they jointly receive is 100.
So to extend the logic, the creditor problem can be rephrased in the following way: which split is suitable, given that every pair of creditors would divide their money by equal division?
The unique answer is the algorithm provided above. Everyone gets an equal amount until they get 1/2 of their claim, at which point they have received as much as they could by equal division of the contested portion. Then, once everyone has 1/2, the money is divided in reverse.
Hope this clarifies matter, but I am fully ready if there are more questions!
By Presh Talwalkar on Aug 27, 2008
No, it does not clarify the matter. I checked the original article - thanks for the reference - and there are some things in it that bug me. Strongly.
Now, let me first state I have nothing against the reasoning in the two contestants cases. The principle of equal division of the contested sum seems preety good as an explanation for these cases. (A side note: Interestingly enough I suppose that the way Salomon addressed the case of the two mothers for one child was by treating it as a contested sum, but I’ll leave this for another moment.)
My problem is with the solution for the three contestants case. It has a very odd definition of “equal”: For Auman “equal” means “equal between two parties”, in other words, it means 50%. But there’s nothing in the case to sustain this reasoning. In fact, quite the contrary. If the estate is 100 then “equal” means 1/3 not 1/2. Aumun’s solution is arbitrary, at least in my reading.
But let me handle it according to your algorythm. My issue is with step 2. Why “one half”? Why not a different proportion? Why not 1/3 since that’s the case if the estate is 100? Why not more thant 1/2? This limit is purely arbitrary and seems to have been chosen, not because it follows a principle, but instead because it gives the results pretended.
By Sergio on Aug 28, 2008
Sergio:
This looks like a case where we might have to agree to disagree. I respect that you read Aumann’s article. I found his logic convincing, but you raise a question that it is just a coincidence and concocted to be consistent.
My reference was for Aumann’s lay person explanation. Apparently there is a journal article with the math on why the division is unique and how it can be applied. I wonder if Aumann has more details there. Nonetheless, even if the Talmud does not contain game theory, I have to admit it is a worthy example that motivated me to learn more about coalitional game theory, and that is a triumph of Aumann’s creativity.
By Presh Talwalkar on Aug 28, 2008
Presh:
I enjoyed my progression of understanding of this article. I initially realized the 200 solution shows an even division of 150 for the three creditors, but I didn’t realize that I didn’t understand why only 150 was split among the three, while 50 was split between the two. I then came to your algorithm and it all clicked. Fantastic. Why didn’t I take any economics classes ever?
By Erik on Aug 29, 2008
Like Tim, when I first read through it didn’t make sense because I also assumed that ‘equal division of contested sums’ applied at all levels (which I’ll call the ‘pure’ version of interpreting that statement for want of a better term). This is partly because in your partial math examples you seemed to use arbitrary amounts (”how would you split 125 between A and B?”; where did that 125 come from when the estate was 200?) However Tim did make an error in his math.
With a 200 estate and claims by A, B and C of 100, 200 and 300, respectively, using the ‘pure’ interpretation the split would be: 100 commonly disputed would become 33.3 each, 100 disputed between B and C would become 50 each, yielding overall payouts of 33.3, 83.3, 83.3 (not 33/66/66).
It’s also apparent that things will get more interesting when you add more creditors. Adding a fourth, D, who is owed 100, the split of a 200 estate using the ‘pure’ interpretation would be 25/75/75/25. Using the Talmudic version, the split would be 50/50/50/50.
Under the ‘pure’ interpretation, the more creditors that have claims, the less value a (relatively) low-end claim has. Suppose there were 10 creditors, 9 claiming 100 and 1 claiming 200. Under a ‘pure’ interpretation, splitting an estate of 200 would have the 9 creditors get 10 each while the 200 creditor got 110 — 11 times as much. Under the Talmudic version all 10 would receive 20. In that case it would seem pretty easy to see that most of the people involved would deem it an unfair split; most received just 10% of their claim, while one person recieved over 50% of their claim.
The Talmudic version seems to use 3 basic principals: try to split proceeds evenly; noone gets more than 50% of their claim until -everyone- receives at least 50% of their claim; when everyone has at least 50% of their claim, try to split losses evenly.
In other words, the first stage of the process favors the smaller creditors, while the last stage (if you get there) favors the larger creditors.
In the case of the garment where one person claimed 1/2 and another person claimed the whole, yes, it favors the one claiming more because you’ve gone past what was needed to fulfill 50% of both claims (the total of which would be 75% of the garment, and you’re parcelling out 100%). But add 3 more people claiming 1/2 of the garment (which really is a silly thing to argue over..) and the 5-way split that the Talmudic version yields means everyone gets 20%.
In that case, the difference between being one minority creditor (the original one claiming 1/2) and one of multiple minority creditors is a minor loss — a drop from 25% to 20% — whereas the loss for the majority creditor was significant, going from 75% to 20%.
So the fewer creditors, the better off you are being a majority claimant. The more creditors, the better off you are being a minority claimant. Doing the division using a ‘pure’ interpretation would be moderately in favor of the majority claimant when there are few creditors and overwhelmingly in favor of the majority claimant when there are many (small) creditors. In other words, it wouldn’t scale ‘fairly’ like the Talmudic version does, and would instead encourage the behavior of trying to claim as much as possible. The Talmudic split is such that not claiming as much as possible is not as detrimental to your resulting reward, and thus less likely to encourage greedy behavior.
By David on Aug 30, 2008
Presh, of course my issue is with Aumann’s proposal, not with your understanding of it. In any case, it’s an interesting problem and I only came to know about through your column.
David puts it better than me the issues I have with Aumann’s solution, specially what he states in his first paragraph on the 125 amount. Yet, I still think there’s another arbitrary amount: Why awarding 50% of the claim to each claimant first? There’s nothing in Aumann’s article that sustains this - except for cases with two claimants, of course. But we cannot deduct from a two claimants case a solution for more claimants. After all, the only clear solution for the three claimants case is the first, the one where the estate is 100, and in this case the division basis is not 50%. To interpret “equal” as meaning equal among two people explains nothing.
Note: This is not to say that it is wrong, all I mean is that we may have a working algorythm but we don’t know why it was used.
I didn’t study the issue deeply but there’s something that came to my mind: How good were people at that time with calculus? What tools had they to complex calculus like dividing somenting among many people? I suppose they would be very limited in their ability to perform complex divisions. I’m sure they would favour simpler divisions. Let’s see how this reasoning could help us in our problem.
But before that, a key consideration. I suppose even that many years ago people would have a notion of proportionality and the desirability of spreading an estate in porportion to the claims. Their problem would be how to do it. In the Thalmudic case a proportional division requires dividing the estate by six. It could be hard to do with their mathematical knowledge. (Note, I’m only guessing since I know nothing about the mathematical skills they had.) Thus, instead of doing a single but complex division, it would make more sense to make several simpler divisions.
Still, there’s a notion that should be aparent: Any claimant should get a share that was equal or bigger than the share of a claimant with a smaller claim. I think the examples show this is as far as they went in terms of ensuring some sort of porportional division of the estate.
Furthermore, proportionality is not a primary issue in any case. Consider an estate of 600 and two claims, one of 100 and the other of 200. Both get their claims fulfilled independently and no one cares to see if there’s any sort of proportionality.
In any case, I’ll call limited proportionality (LP principle) the key concept that a claimant with a larger claim should get a share that is equal or bigger than any claimant with a smaller claim.
What’s the simplest division? By one. If there are two claimants to something, one with a claim to the whole, the other with a claim to a part, divide the whole into two parts. The part that is only claimed by one is assigned to him direct, in other words, it is divided by one. But than at this stage the claimant that claimed the whole has already seen his claim fulfilled in part. So, it makes sense at the next stage to give compensation to the other claimant since he is still waiting for compensation.
But since there are two claims to the contested part of the estate, it’s handled with the next simplest division, by two. Give an half to the contestant that is still to get some compensation. Give the other half to the contestant that already got some compensation. Finally, check if the shares each receives respect the LP principle. If they do, stop at that.
This is simple and straightforward with two contestants, but what if there are three? It depends on the case.
Case one, all have a claim to the whole estate (first of Aumann’s cases). There’s no way to handle it through a division by 1 or a division by 2. One has to go for a division by 3.
Case two, one has a claim to a part of the estate and two have a claim to the whole (second of Almann’s cases). Divide the estate in two shares, one that is claimed by 3 - the more contested one - and one that is claimed by 2 - the less contested one.
Divide the less contested share among the two contestants.
Divide the more contested share in two. Assign one half to the contestant that is still to receive some compensation.
Divide the other half by the contestants that already received some compensation, and sum their respective shares from the two parts.
Check if the shares respect the LP principle. If they do, that’s all. If they don’t, revise the whole division procedure by dividing the estate by three.
Finally, we have the last of Aumann’s cases. The way to handle if follows a similar procedure.
Divide the estate into three parts, each one corresponding to different degrees of contestation.
Pick the less contested part and assign it to the single contestant. At this state he got some compensation, so the next stage will try to ensure that the two other contestants also receive some compensation.
Divide the more contested part by the two contestants that are still to receive some compensation, half for each. At the end of this stage all contestants received some compensation.
Now it is time to finish the division by dividing the third part by its contestants.
Check if the LP principle is respected.
As you can see, I reach the same results that Aumann reached, only using a different algorythm. I also think that I explain the reason for dividing the estate parcels by two, something that Aumann does not explain. As far as I recall, I don’t have recourse to game theory, though. Instead, I base my approach on the possible limitations in terms of knowledge of mathematics and computation.
Needless to say, I’m only brainstorming.
By Sergio on Aug 30, 2008
i can understand the mechanisms but i can’t think of a single situation in wich fair & smart ppl would prefer it over proportional distribution.
By Nicola on Aug 30, 2008
Doesn’t all of this start with a base-line assumption that fiduciary value is fungible?
The math looks good and it certainly provides an algorithm for interpreting the puzzle laid out in the Talmud. However, as far as I understand economics (this could be the problem), the value of a given object is not fixed to mathematical constants such as those provided in the above examples.
A real-life distribution system based on the “equal distribution of contested sums” would prescribe certain values to items that may not accurately express the value ascribed to that item by, oh say, the laws of supply and demand, or even downright human sentimentality.
The example of the garment illustrates my point rather vividly. If there are several claimants, be they minority or majority, the garment is subjected to division. And as only owning 20% of
the garment provides insubstantial and therefore unusable clothing, couldn’t the value received as interpreted by the claimant said to be 0%? In fact, in the case of the garment any division would cause this outcome, so then the proportional difference between the the minority and majority claimants is also effectively driven to 0%.
I’m just wracking my brain to come up with a situation in which the “equal distribution” scheme doesn’t eventually encounter the problem of subjective values. Even in the case of currency value, where there are nice round numbers to divide, instead of lands/cows/grain/etc, aren’t their still all sorts of factors other than the number of claimants and the size of their respective claims which go into determining value?
Take the first example of the 3 wives and their 100/200/300 split. If the 100wife has remarried an can instantly re-invest her share in her new husband’s farming venture, whereas the 300wife has moved provinces and has to exchange her inheritence into a new currency, What can then be said about the proportionality of the distribution?
It seems that the data feeding the examples provided is strictly dependent on an outside, pre-arranged prescription of both the original value of the estate and it value once divided, such as a marriage contract or will.
I realize that this article does not set out to justify this scheme of “equal distribution” but rather to divine the knowledge of the wise old scholars who brought us this Talmudic brain teaser. For that, I stand in awe and applaud. However I still find this a deeply unsettling world view and had to raise a point or two in the defense of sanity.
Cheers
By Sanjian on Sep 1, 2008
This might be easier to see if you think in terms of you owing creditors, instead of you receiving money. For example: you have 4 debts: $63.00, $99.50, $1,389.00 and $12,000.00. And you have $200.00 per month to pay these debts after living expenses are taken care of. You can use this system to set payments.
1st month Pay Still Owe
Debt 1 63.00 31.50 31.50
Debt 2 99.00 49.75 49.75
Debt 3 1,389.00 59.37 1,329.63
Debt 4 12,000.00 59.38 11,940.62
2nd month Pay Still Owe
Debt 1 31.50 31.50 0.00
Debt 2 49.75 49.75 0.00
Debt 3 1,329.63 59.37 1,270.26
Debt 4 11,940.62 59.38 11,881.24
3rd month Pay Still Owe
Debt 3 1,270.26 100.00 1,170.26
Debt 4 11,881.24 100.00 11,781.24
etc. Now lets say you get a windfall of $10,000 after paying down your debts for 5 more months. Recalculating with $10,000 + the $200 lets you pay:
9th month Pay Still Owe
Debt 3 770.26 385.13 385.13
Debt 4 11,381.24 9,814.87 1,566.37
Recalculating back to $200 a month:
10th month Pay Still Owe
Debt 3 385.13 100.00 285.13
Debt 4 1,566.37 100.00 1,466.37
and so on.
So, who’s going to get a windfall of $10,000? Perhaps a more realistic value is a $850.00 tax return. The point is, you can use this system to pay off debts without bankruptcy nor “credit management” companies. Most credit cards and hospital bills can be settled in full if you offer 50%. This can help you get rid of the smaller debts and set payments for the larger ones.
That’s my $0.02 worth, Steve
By Steven Miller on Sep 1, 2008
Wow, these comments are amazing and beyond my wildest dreams. I’m going to send these to anyone that thinks people don’t care about economics or math!
I appreciate all the thoughts. Some of you have made me realize my prose was unclear so I have revised the article to be more precise in my intention.
I have run out of ideas for this topic, but lo and behold, there are other people who wrote about this problem in a very interesting way. Check out this Slate article for another explanation. It made me realize that consistency was a big issue (reflected in my edited version of the article):
http://www.slate.com/id/2047/
By Presh Talwalkar on Sep 4, 2008