. It was originally due to the third phase of data streaming 16th.
At that time, I was teaching at the American Buddhist University.
. Reprinted from: http://episte.math.ntu.edu.tw/articles/mm/mm_16_3_08/index.html
Thank you for guiding: http://blog.codingnow.com/2006/04/shapley.html
| Preface |
"An employee's work is desperate, and the old employee says nothing. The employee's work is profitable and has the right to share the work, <strong margin> In the long article, the employees demanded fair treatment from their families. The Dynamic Moments tell us some social injustice. "Injustice": not only do the victims feel the same, but the neighbors are also angry and tolerant, and the benevolent and miserable, who are eager to call for help, for a wise man, how can we decide whether to be fair? Is it because the cost of the company has been reduced too much, or has the employee requested a copy? You may not believe in a simple mathematical formula to understand this problem, and let us see a new example, understand the truth of man.
Some people have said that the question is the soul of mathematics. Let's start with the question. Assume that there is one database owner, one engineering engineer, and two workers. I have paid for my work, but I cannot pay for it if I have no project or workers. If the company cooperates with the engineering team and there are no workers, the two will be used for a short time. To do some work for the workers, they can earn three yuan a month. If a worker is added, the workers have the ability to improve their skills, the efficiency has increased dramatically. The three-person cooperation can earn RMB 6 per month. If a new employee is added, RMB 9 per month. However, if only the primary and workers do not have a Project License, the job cannot start. Now, how can we make the formula fair and reasonable for these 9 RMB profits? (Please note that these nine yuan are completely profitable, and the work guarantee, information interest, and shareholder benefits have been deducted .)
The primary may say to the worker: let's add five to each other. Let's give you one to five in a month.
The worker may answer that this is not too dark. Every month, each of us will "renew" for three months. Why can't I give them one person? In this way, you can still be allocated to the two-party profit center of our labor center. Why can't I renew it.
The speaker will say: I will not give you a job, I will not give you a job, I will say I will never give you a job, or I will give you a five-day tour! As a result, the employee started to work properly, and the master said the master was justified. So clearly you will be standing there? Can we solve this issue rationally? Before reading this article, I hope that you can combine this article and think about how to solve this problem. First, I want to figure out the answer, which is an important way to increase the interest and skill of data.
Thought about it? So I will tell you the answer first. A fair distribution should involve three yuan each for the primary and engineering workers, and one yuan for each of the two workers. Unfair? No, there will be no more unfair law.
Before moving to data, we can look at the breadth and depth of this question. Almost all economic issues are related to this issue. The primary school has made a lot of money, and it is essential to generate a large number of employees, experts, technicians, and police officers. How should they allocate this huge amount of money? What are the differences in the treatment between the Senior Colonel, faculty member, staff member, and staff member of the school? How much is the Professor's contribution to the academic discussion? Further, as a member of the community, we can do the same for how much we should pay and how much we should pay. The foundation of all rights should also start with fairness. I do not treat others, but I do not allow others to treat me.
| Shapley principles of fairness |
First, we must understand the issue of unfair distribution of public networks only when cooperation is available. Otherwise, when you go through your website, I will catch them, eat mountains in the mountains, and eat water by water, in this case, the problem of uneven distribution is not found. Today, ω is a job that can be operated by multiple people.SA collection of people who can join the job. IfUForSA subset.UBenefits of human cooperation. For example
S =/{/mbox {/fontfamily {cwM1}/fontseries {m}/selectfont/ch...
IfU= {Primary, project cost}, (cost yuan), ifU= {Primary, engineering engineer, worker a}, rule, ifU= {Primary, worker a}. As a result, we can see it again.STo 1, 2 ,...,NFor the token, that is. MeaningIMembers should be rewarded for work at ω. In his comments in 1953, jublel laid three principles of fairness.
No one will disagree with this principle, that is, the equivalent operator. If the third party can do the fourth party, the third party can get the reward of the fourth party, it has nothing to do with his name: James or James.
This is also a fair principle,UThe benefits of a person are naturally dividedUPeople in.
This is also the right principle. I do two jobs and I can earn two reward points.
It is rare that (he is also an odd man in his original post ). You can find the three principles.
/Phi_ I (/Omega) =/SUM _ {u/subseteq s} R_N (s)
S=UElement number,
AndU-{I} indicatesUMember {I} is removed from the worker }.
Although the proof of this theorem is not long (about two degrees, about the high middle number), we do not repeat it here, those who are interested in this article can find them from the transcript of this article. In addition, we will perform a direct analysis on the (1) type in the following text. Even if no evidence is found, many people will believe that (1) is a fair allocation method. To understand the usage of this formula, let us calculate the reward of the primary account.
S= {1, 2, 3, 4}
1, 2, 3, and 4 represent the primary, engineering, and worker A and B.UYesSA total of 24 = 16 child sets. The optional columns are as follows:
R_2 =/frac {(2-1 )! (4-2 )!} {4 !} =/Frac {1} {12}/mbox {/fontfamily {cwM0}/fontseries {m}/selectfont/char 1 }}
&/Omega (1, 2, 3)-/omega (2, 3) = 6-0 = 6 ,//
And
R_3 =/frac {(3-1 )! (4-3 )!} {4 !} =/Frac {1} {12}
/Omega (1, 2, 3, 4)-/omega (2, 3, 4)
Therefore
/PHI _ {1} (/Omega) & = &/frac {1} {12}/times 3 +/frac {1} {12}/times
Likewise
/Phi_2 (/Omega) & = & 3.5 //
Now, it is time to say that this allocation is unfair, because once you agree to the three principles of balancer, the rest of the tweets are not pushed down, likeA+B=B+AThe mathematical principle of this type. While the original law of jublel seems to be lax. Therefore, we have to admit that this 3.5: 3.5: 1: 1 is a fair allocation method.
From another perspective, we found that
/Omega (U)-/omega (U-/{I /})
Indicates that {I} is added.UThe increase (I}) advantage, that is,U. WhileRN(SIn all 1, 2 ,...,NArranging, {I} inSPosition to maintain the probability that the previous and subsequent members do not change, becauseSYesUThe element number, that is, {I} isUThe probability of adding the nearest member. Therefore, (1) represents {I} membersSThis is the reward for {I. It seems that it is not easy to make it clear and let's look at an example. Because limit 4! = 24 is a little too big. We have one fewer worker in our workshop to see what the allocation is. At the same time, it can also affect the income of each person when one worker is less than a worker. NowS= {1, 2, 3} represents the primary, engineering, and worker a in turn. On the left side of Table 1, all three numbers are arranged. The second, third, and fourth columns represent the profit margin brought by each person in this sort. Take the first row as an example. When the primary node arrives first, he cannot start work. Therefore, the benefit brought by the primary node is 0, and the second node of the Project arrives. He and the primary node can work with the primary node to get 3 benefits, therefore, he brought 3 (not all of his achievements, but he brought them), and the workers finally came, he brought three more profits, so these three records are recorded in the workers' names. Therefore, in the first line, the profits brought by each person are 0, 3, but we have no reason for the primary to come first! There should be equal opportunities (fair, isn't it ?) Therefore, on average, the profits brought by each person are 2.5, 2.5, and 1. The last three columns of Table 1 represent the relationship between the table and the (1) formula. As long as the child accounts take a picture of the relationship, we can see that (1) it indicates the average value generated under these orders.
| Arrange | 1 | 2 | 3 | Containing 1U | S | |
| 123 | 0 | 3 | 3 | {1} | ||
| 132 | 0 | 6 | 0 | {1} | 1 | 2/6 |
| 213 | 3 | 0 | 3 | {2, 1} | 2 | 1/6 |
| 312 | 0 | 6 | 0 | {3, 1} | 2 | 1/6 |
| 231 | 6 | 0 | 0 | {1, 2, 3} | ||
| 321 | 6 | 0 | 0 | {1, 2, 3} | 3 | 2/6 |
| And | 15 | 15 | 6 | 1 | ||
| Average (and/6) | 2.5 | 2.5 | 1 |
Table 1: comparative advantage for {I}
If we compare the result of table 1 with the result of two previous workers, we will find that many workers B and former workers a share equal pay for equal work (reasonable, ?), I got a bonus, but the other two benefits he caused were swallowed up by the hacker and the project. (do you think the workers are unlucky? There are even more unfortunate things in the backend, if it is really fair .) Therefore, without being pushed by the three principles, or by the (1) method, we find that fair distribution is not the same as what we think. He can see that this point has led him to become a generation of clan, and his formula has become a column stone of mathematics and economics.
| Interesting results of the Gini Theorem |
Work cannot increase profits, so his reward is 0. This person is a dummy employee becauseIThe limit is 0.
In this topic, we can see that the project team and the Primary Team share the same reward, which seems to be different from the actual situation. In most jobs, the project team cannot receive the primary team's treatment. This is because in general, there are more than one project, especially when the project is provided for the sake of demand, their bodies will fall. Current SettingsS= {Primary master, Project A, Project B, and worker a} their tokens are still 1, 2, 4, and 3 in sequence. However, only one project is needed. At this time, if the primary Party asks one of them, he will still pay him two yuan, five yuan? See the result in formula (1.
The formula (1) is wonderful, but unfortunately (fortunately) in general, it is not easy to calculate because it containsSAll subsets in, with 2N. WhenNWhen the computing volume is a little large, the computer will die. However, in some cases, when the Members have the same abilities, the arrangement method in Table 1 can be as small as possible. Here is an example.
Ask a businessman to ask a person in a village to renew his account. At the time of receiving the token, he announced that he could only copy the token to the correct account, with a price of one thousand yuan. When the village staff gathered together, they found that there were 110 million male users and 90 million female users, so they could earn RMB 9 million, in order not to let Chen xiongjia's home have broken his head, the whole village was angry, so everyone had a total of 9 yuan, and 20 yuan was released. The question now is how to make money fair 」. When Chen xiongjia's home is hosted in the primary region, that is, each region has a value of 9 RMB/200 = 450 yuan, but the female thinks that things are thin, the reason why female live less is that it must be a bit more efficient than just getting started again. How can this problem be solved? If we look at fairness based on the principle of 'baler', we can find out that there are only two types of fairness.
/Phi _ {I} (/omega) =/sum _ {k = 1} ^ {n-1} r_n (k + 1) /sum _ {x/geq [/frac {k} {2} + 1]} ^ {k} {n_2/choose x} {n_1-1/choose k-x}
Formula [Y] TableYThe integer value,N1IThe number of records suchITake the leadN1 = 110,IFor femaleN1 = 90,N2. Another kind of data,N=N1 +N2. The result calculated by the computer is that the value of the male ticket is 109 yuan, and that of the female ticket is 876 yuan. The value of all male tickets is only one of the 6.5 points of the female ticket. In fact, there is a slight imbalance, and the gap is very poor. If the male and female are respectively 102 and 98, the ratio of the unit is. 82. Things are thin and thin, consistent with this.
I remember a few years ago when I was married, there were many male and fewer female, and there were hundreds of times more girls. Nowadays, it seems that there are more female and fewer male, and the market is very simple. However, we cannot treat every young boy or girl as a teenager. Because each person has different terms, it is difficult to calculate the fair status of each person from (1) (2NSubset while ?), A good combination of men and women does not mind how much they love. However, the (1) formula can at least resolve an important phenomenon. If it is not balanced, a large margin is generated.
Actually, people do not know the formula (1), but this phenomenon has already been deeply rooted in people's hearts. Everyone wants to change their self-esteem, making the formula (1) it is good for me. When the food, milk, and wool were poured into the sea to increase the quota. For example, if the male family first discusses the matter and secretly releases the 20 rule, it will be advantageous to the male family. (If the 30 rule is used, it will be more advantageous, at this time, the value of female is only 201 yuan per region, while that of male is 773 yuan. In this case, only 80 yuan is left for the family, but it is favorable for the family. However, cooperation is required. If you put the token on your own, you may not be able to get one article .) Various Industries prevent users from being used to limit the number of users by using licenses, conferences, support meetings, and headers. Everyone knows that they want to become an indispensable person in a business.
However, some people do not like injustice, but they want to break through fairness. Some people break through the principle of first place and use various relationships to make social media have a relationship with their names, he is my son, and his reward will naturally increase. Some people break down the second principle, and those who do not do things will create their names and make some money. Of course, the most important thing is that people who have insufficient production power break down the fair distribution.
| Thoughts |
From the perspective of fairness, it seems that the talented people in the community have not yet obtained fair social payments from them. These indispensable entrepreneurs have invented their families, according to the schibler formula, their reward may be unacceptable. So when we do not complain for ourselves again, we should think of the formula (1): from a fair perspective, we are the one who cares about me, or am I a global speaker?
The Shapley value: Essays in honor of Lloyd S. Shapley, 1988, ed. by a.e. Roth, combridge University Press.
Shapley's 1953 discussion is also included in this article.
Start building with 50+ products and up to 12 months usage for Elastic Compute Service
1 on 1 presale consultation
24/7 Technical Support 6 Free Tickets per Quarter Faster Response
Alibaba Cloud offers highly flexible support services tailored to meet your exact needs.
A comprehensive suite of global cloud computing services to power your business
Payment Methods We Support
