AMAZING GAME THEORY post by Presh Talwalkar

How Game Theory Solved a Religious Mystery

image source: marvinxsteadfast via flickr
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 (like my time-tested method for dividing bills fairly in large groups).
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 (full paper). For this article, 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:

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

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.
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:

1. Order the creditors from lowest to highest claims.
2. Divide the estate equally among all parties until the lowest creditor receives one half of the claim.
3. Divide the estate equally among all parties except the lowest creditor until the next lowest creditor receives one half of the claim.
4. Proceed until each creditor has reached one-half of the original claim.
5. 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.
6. Then divide the estate equally among the highest creditors until the loss of the highest creditors equals the loss of the next highest.
7. 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.