Gaming Religion: Ancient Debts Repaid
Despite the horrendous title, this post is not, in fact, about getting the most spiritual enlightenment possible from your weekly worship. Instead, I’d like to take some time to explain an ancient method of debt collecting laid out in the Mishna, a Jewish text written around 200 CE, and part of a collection of works known as the Talmud. The problem presented is one in which a man passes away, owing debts of 100, 200 and 300 (monetary units), but not having enough money to pay all three debts off. Modern “logic” would suggest two solutions: divide it by person, i.e. simply give a third of the deceased’s estate to each person, or divide it by monetary unit, i.e. giving each person a proportional sum based on what was owed to them. Examples of these two “strategies” for a 200-unit estate would be as follows: on a per person basis, each of the creditors would receive 66 2/3 units, while on a per unit basis, the creditors would receive 33 1/3, 66 2/3 and 100. According to the Mishna, however, the debt should be divided 50, 75 and 75. “What?” you may ask yourself. Why would the two highest creditors receive the same sum? And it gets even weirder when you look at more examples, as illustrated in the table below, borrowed from the article which I will comment on later.
Well, it seems like they are just using the first strategy for low sums, the second strategy for high sums, and then some sort of mutant hybrid of the two, right? Wrong. In comes Robert Aumann, an Israeli-American mathematician to the rescue. This method of debt distribution had, until his solution in the 1980s, been largely unsolved, and thought to be either a transcription error, a strange special case, or just gibberish. And how, after nearly 2000 years, did a lowly mathematician break the code? Game Theory.
To begin, Robert Aumann and his colleague Michael Maschler studied another part of the text, where two parties contest a garment. One claims half of the garment, while the other claims the whole thing. Using our two “modern strategies,” we would either give half to one and half to the other, or give 1/3 to the half claimant and 2/3 to the other, as would be proportional. However, the Mishna claims that it should be divided ¼ and ¾. It is hear the scholar introduces the concept of “equal division of the contested sum.” The logic goes like this: the whole garment isn’t what is being argued over, it’s half the garment. So, that half of the garment is split between the two parties, and the party claiming the whole garment receives the uncontested half, resulting in the aforementioned ¼ ¾ split.
Now that’s nice and all, but how do you do that with three creditors? Pull any two from our handy table above, and the “equal division of the contested sum” holds. Say you take creditor 100 and 300 from estate 200. One gets 50, the other gets 75. Their total share is 125. The contested sum is 100, so each get 50, then the remaining 25 goes to the one with the higher claim, totally 75, which is consistent with the table. Aumann makes the conjecture: “The division of the estate among the three creditors is such that any two of them divide the sum they together receive, according to the principle of equal division of the contested sum.” He also says and shows that neither of our “modern” solutions hold true to this claim, as well as any other solution, making the table unique in it’s solutions.
Aumann then goes on to explain expanding the two claimants case to higher amounts, including our original 3, but does a weird visual proof, which would be difficult to describe in a reasonable amount of time (or an even less reasonable amount of time than what I have used). Instead, I will show an algorithm from the an entry in the blog “Mind Your Decisions.”
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.
Going through this algorithm can be quite fun, and I would highly suggest trying it with 3, 4 or even 5 creditors and a variety of estate sizes.
What is interesting to me about this problem is that it originated from clearly very different social values. As I mentioned, there would seem to be two distinct solutions to dividing of sums, whether it be splitting a dinner check, or rationing food. The strategy of “equal division of the contested sum” seems quite foreign to our contemporary understanding of such problems, but apparently was apparent and reasonable to Jewish scholars of the third century. So maybe I wasn’t able to help you reach Nirvana, but next time you have to decide how much of your dead relatives land you get, maybe try something new for a change?
Sources: http://dept.econ.yorku.ca/%7Ejros/docs/AumannGame.pdf
http://mindyourdecisions.com/blog/2008/06/10/how-game-theory-solved-a-religious-mystery/#.VBySsUuppFw