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A story about Jewish inheritance that has been passed down for thousands of years. The result may seem contradictory, but there is a distribution principle that runs throughout. The paper by Nobel Prize winner Robert Auman solved this ancient mystery and proved for the first time from the perspective of modern game theory that the ancient Jewish rulings were completely consistent with the principles of modern game theory.

About the author: Zhang Ping, Ph.D., has taught at Peking University, the Hebrew University of Jerusalem, Israel, and now teaches at Tel Aviv University, Israel

Among ancient Jews, they were proficient in the law. The scribes were called "rabbis". Rabbis not only studied Jewish law, but also served as judges in civil courts and adjudicated civil cases. During the Talmud era (1st to 6th century AD), rabbis already had excellent knowledge of game theory. A paper published by Nobel Prize winner Robert Auman in 1985 revealed the wisdom of ancient Jews in solving fairness problems from a short story.

"Talmud·Women's Department·Marriage Book"

If a person dies after marrying 3 wives, the wife's marriage book (compensation for the termination of marriage as stipulated) is 1 ma (equal to 100 zi), that wife's is 200 zi, and the other wife's is 300 zi, and there is only 1 manai, they will share it equally; if there are 200 zi, then the one with 1 manai will take it. Those with 50 sets, 200 sets and 300 sets each take 3 gold dinars (1 gold dinar is equal to 25 sets); if there are 300 sets, then those with 1 mane take 50 sets, 200 Those who make up the gift will get 1 manai, and those who make up 300 will get 6 gold dinars. Similarly, if 3 people put money in the same purse (partnership in business), losses or profits will be distributed accordingly.

The mysterious "marriage letter"

The "marriage letter" is a letter written by an ancient Jewish man to his wife when he got married. One of the important contents above is in case the marriage is terminated (death or divorce) , how much money the husband will compensate his wife. A property dispute is recorded in Chapter 10, Section 4 of "Talmud·Women's Department·Marriage Book". In this case, a rich man promised his three wives in his marriage letter that after his death, he would give the eldest wife 100 gold coins, the second wife 200 gold coins, and the younger wife 300 gold coins (for simplicity, the coins were all changed into gold coins. ). But when people liquidated his estate after his death, it was discovered that the rich man had lied. His property was not 600 yuan, but only 100, 200 or 300 yuan. So, how many gold coins should each of his three wives get at this time? The property distribution plan prescribed by the rabbis (referred to as the "Talmud Plan") is shown in Table 1.

According to common logic, there is obviously a serious problem with this form. Because the ratio of the inheritance that these three people deserve is 1:2:3, and in the rabbis' ruling, this ratio is only true when the amount of the inheritance is 300 gold coins. Many Jewish scholars have seen this contradiction very early. As for why this contradiction occurs, and whether there is a distribution principle that runs through these distribution methods, no one can give a reasonable explanation. An eternal mystery.

The mystery was not solved until 1985, when Robert Auman and another scientist published a paper titled "A Game-Theoretical Analysis of a Bankruptcy Problem in the Talmud" . This paper proves for the first time from the perspective of modern game theory that the rulings of the ancient Jewish rabbis are fully consistent with the principles of modern game theory. Since then, the story of "three concubines competing for property" in the Talmud has become one of the earliest examples of human understanding of game theory. The first key to unlocking this mystery is actually still in the Talmud.

Chapter 1, Section 1 of the "Talmud Damage Section" provides the following resolution principles for both parties in property conflicts:

Two people grabbed a coat. Said, this is what I discovered; the other said, this is what I discovered. One said, this is all mine; the other said, this is all mine. Then this person must swear that he owns not less than half, and that person must swear that he owns not less than half, and then they will be divided equally. If one says, this is all mine; the other says, half of this is mine. It means that all owners must swear that they own no less than three-quarters of it, and those who own half of it must swear that they own no less than one-quarter. The former will take three-quarters, and the latter will take one-fourth. one.

What the Talmud proposes is an unusual principle for resolving property disputes. This principle is called the "dispute coat principle." This principle mainly includes the following two contents:

1. The disputing parties shall only allocate the disputed part, not the undisputed part. So the person who claimed half the coat would first lose half the coat and would have to share half the coat with the person claiming all the coats.

2. In a dispute, the person who makes a higher claim shall not receive less than the person who makes a lower claim.

The contribution of Robert Auman's paper is to find the connection between these two paragraphs. After studying these two passages, the paper proposes the following theorem:

The Talmudic solution is the only solution consistent with the principle of the coat of contention.

Take the issue of property disputes between three concubines as an example. According to the Talmud plan: when the inheritance is only 100 gold coins, the three concubines have the same right to claim the entire inheritance, so the three concubines’ equal share is consistent with the "dispute" The coat principle”.

In the Talmud plan, the property distribution result between any two of the three concubines also conforms to the principle of dispute overcoat. When the number of inheritance gold coins is 200 pieces, the eldest wife and the second wife get 125 pieces (equivalent to two people fighting for 125 pieces). Since the eldest wife can only get up to 100 pieces, the second wife gets 25 pieces first. Since both of them have the right to get the whole remaining 100 yuan, they are divided equally according to the principle of dispute overcoat. In this way, the eldest wife gets 50 yuan and the second wife gets 75 yuan. At this time, the property distribution results between the two people are consistent with the principle of the coat dispute.

In the case of an inheritance of 300 yuan, the eldest wife and the second wife compete for 150 yuan. Based on the same principle, the second wife gets 50 yuan first, and then they share the remaining 100 yuan equally. In this way, the eldest wife gets 50 yuan, and the second wife gets 100 yuan.

Even better, the Talmudic solution not only ensures that the gains of any two people in the distribution of property are consistent with the quarreling coat principle, but that the losses of any two people are also consistent with this principle. When the inheritance is 200 yuan, the second wife deserves 200 yuan, the actual income is 75 yuan, and the loss is 125 yuan. The younger wife loses 225 yuan, and the second wife and younger wife together lose 350 yuan. According to the dispute principle, since the second wife’s request is 200 yuan, the younger wife will lose 150 yuan first. At the same time, since the younger wife’s request is 300 yuan, the second wife will also lose 50 yuan. This leaves only a loss of 150 yuan, which is divided equally between the two, with each losing 75 yuan. In total, the second wife loses 125 yuan and the younger wife loses 225 yuan.

How to distribute, an eternal question

The "Marriage Book" only specified the distribution plan, but there was no calculation method in the original text and annotations, so it became an eternal mystery. Experts speculate that there are two ways to calculate the Talmud solution.

Method one is very simple, which is to divide the property equally by dividing the total number of property by the number of people sharing the property.

Method 2 is a little more complicated. First find the person with the least requirements (we call it the first one), then treat the rest as a group, and make the first allocation between the two parties. . Since anyone in the group has higher requirements than the first person, if

the allocation between the first person and the group complies with the dispute coat principle, then the allocation between him and anyone in the group will also be should comply with this principle. The income is then distributed among the group members for the second and third time using the same method, and so on.

Specific to the story of "Three Concubines Fighting for Property", when the number of gold coins inherited is 200 yuan, the eldest wife and the second wife and the younger wife group make the first distribution. Since the eldest wife only gets 100 yuan, the second wife and the younger wife group get 200-100=100 yuan first. The remaining 100 yuan is divided equally between the two parties, the eldest wife gets 50 yuan, and the second wife and the small wife group get another 50 yuan. In the second distribution, the second wife and the younger wife had full claim to the 150 yuan they received in the first distribution, so they split it equally, each receiving 75 yuan.

It should be said that method two is the basic calculation method of the Talmud plan, but there is a limit, that is, the result calculated by this method cannot be that the party who asks for less gets more than the party who asks for more. . If this happens, you need to switch to method one and divide equally. Specific to the story of "Three Concubines Fighting for Property", the limit point is 150. If the number is less than this number, method one must be used. For example, the inheritance amount is 149 yuan. If we don't use method 2, the second wife and the younger wife will share 99 yuan equally, and each person will get less than 50 yuan. This violates the principle of dispute.

The Game of Wisdom

Now let’s take a look at what will happen if the Talmud plan is applied to bankruptcy settlement disputes in real society. In order to facilitate work and comparison, we use the common proportion calculation method to make a comparison.

Suppose A owes B 70 yuan and owes C 30 yuan. Now A is bankrupt. According to the amount of A’s remaining property, using the Talmudic scheme and proportional calculation method, we can get Table 2.

Table 2

A’s remaining property

(yuan) Talmud solution ratio calculation method

B’s amount (yuan) C’s amount (yuan) B’s amount (yuan) B’s amount (yuan)

90 65 25 63 27

80 60 20 56 24

70 55 15 49 21

60 45 15 42 18

50 35 15 35 15

40 25 15 28 12

30 15 15 21 9

20 10 10 14 6

10 5 5 7 3

Here, 50 yuan is a dividing line. On this dividing line, The Talmudic scheme yields the same result as the proportional calculation method. Above this line, B's profit in the Talmud scheme is higher than the proportional calculation method; below this line, B's profit in the Talmud scheme is lower than the proportional calculation method. The situation of C is exactly the opposite.

Now suppose A is a supermarket chain, B is a large food company, and C is a small bakery. Multiply the relevant numbers by 1,000 and we can get a realistic picture. Since bankruptcy is the consequence of severe insolvency, it is difficult to see a situation above the 50 threshold. And when the situation below 50 occurs, the Talmud plan better protects the basic interests of small households than the proportional calculation method. For a large food company, less debt recovery will probably mean less profit; while for a small bakery, a proportional bankruptcy settlement may mean the closure of the bakery. This is what we often see in real life. situation. When a commercial enterprise fails, it is not the large suppliers but the small and medium-sized enterprises that are hardest hit. And if these small and medium-sized enterprises collapse in a chain, the economy of the entire region will be negatively affected. Therefore, protecting the interests of these small and medium-sized enterprises in bankruptcy settlement is a key link, which is also one of the values ??of the Talmud plan.

In fact, the real beauty of the Talmud plan is that it protects the interests of the weak while still maintaining the fairness of the game rules. From the perspective of the entire bankruptcy settlement game, if the Talmud solution rules are applied, then both large and small households have a chance to win, and at least in theory, the chance of winning for both sides is 50 to 50. If the number of assets exceeds half of the liabilities, the big owner wins, otherwise the small owner wins. This kind of impartiality can ensure to a large extent that players on all sides respect the rules.

From a game theory perspective, the Talmudic solution provides an excellent solution to the bankruptcy dispute and is characterized by a consistent principle throughout. Once this principle is accepted, any two parties in a dispute will find that the solution is just from any angle and will not be dissatisfied. Among the various solutions to bankruptcy disputes that modern game theory can provide, the Talmud solution is closest to the "nucleolus" concept of game theory. Therefore, some people say that the Talmud solution is the "nucleolus" of modern game theory. The originator of the concept of benevolence.

Robert Auman won the Nobel Prize in Economics in 2005, of course not because of his paper, but he reminded us of the wisdom of ancient Jews in solving fairness problems.