恭贺新春，虎年吉祥

大家好，这里是Leanringyard学院，今天是大年初二，小编在这里祝大家新的一年万事顺遂，虎虎生威，红包多多。在这充满欢乐节日气氛的时刻，小编也不会忘记给大家带来新知识，所以今天继续为大家带来Mathematica入门教程系列文章，今天的主题就是：Shapley值法。

内容提要

上一期，我对《电商平台扣点率影响下的双渠道供应链协调定价研究》（作者：张伸，孟庆春，安国政）中使用的理论，陌生名词，概念与论文构建模型的思路与方法进行了梳理，在结尾的时候提到了协调方案，文中用的是基于shapley值的协调方案，而小编也是第一次知道这个方法，所以就在前几天进行了简单的学习，这一期我们就来看看什么是Shapley值，它又是怎样在文中进行运用。

Shapley值法

本来想百度一下，但百度说的有点不清楚，所以还是去知网找了相关论文进行学习，马士华,王鹏的《基于Shapley值法的供应链合作伙伴间收益分配机制》，这篇文章对shapley值法定义进行了详细的描述，并通过一个简单具体的数值算例展示了如何在供应链协调中如何使用。

1.目的：基于我自己的理解就是有多个参与者从事某项经济活动，在最优组合策略下，这活动将产生一个最大总效益，他们将采取最优的分配方案使得自己的效益最大，这一分配方案就是SHapley值法。

2.定义：

上图则说明了Shapley值法的定义，首先要满足条件：没有成员时，收入为零；总体合理性与个体合理性。然后，列出可能的联盟集合。最后根据公式计算各成员分配的收入。

在参数方面并没有做过多的解释，所以我有参考了另外一篇文章，并对每一个公式做了说明：

（1）：表示在没有企业时，整个活动收入为0；

（2）：表示超可加性，类似1+1>=2，说明企业联盟时的总收入要大于等于企业不联盟时各自收入之和；

（3）：表示总体合理性，各成员从联盟的总收入分配到的收入之和等于联盟总收入；

（4）：表示个体合理性，个体从联盟中分配到的收入不能低于独自完成的收入。

（5）：表示成员收入，Shapley所给出的局中人i的收入称作 Shapley值；

（6）：权重最终计算公式。

这就是整个Shapley值法模型，证明过程小编就不展开说明了。

实际运用

知道了shapley值法的基本原理之后，我们再来看看它在文中的具体应用。在文中的供应链中只包含制造商和零售商两个决策者，先确确定制造商分配的收入，联盟集合I={制造商，（制造商，零售商）}。先根据收入计算公式，确定各参数具体数值，如表所示，然后计算结果，下图表格中最后一排则表示制造商在每一种联盟下的能够分配到的收入。

将两种联盟下分配的收入相加则表示协调之后的制造商利润。同理可以求出零售商的利润。

确定协调之后的利润表达式之后，令其等于分散决策时的利润表达式，在使用软件计算出对应的批发价格即可。

END

本期分享就到此结束了，我们已经把文章的全部理论知识以及建模思路与方法学习清楚了，下一期就开始使用Mathematica进行实战演练了，让我们期待一下吧。对于Shapley值法有兴趣的可以自己去搜索更多的资料学习，也欢迎留言和小编交流！最后再次祝大家新年快乐，新的一年得到新的收获！

参考资料：

[1]谷歌翻译、百度百科；

[2]张伸, 孟庆春, 安国政. 电商平台扣点率影响下的双渠道供应链协调定价研究[J]. 中国管理科学, 2019, 27(10): 44-55.

[3]马士华,王鹏. 基于Shapley值法的供应链合作伙伴间收益分配机制[J]. 工业工程与管理,2006,(04):43-45+49.

[4]刘浪,唐海军,陈仲君. Shapley值在动态联盟利益分配博弈分析中的应用[J]. 工业工程,2006,(06):118-121

部分资料图片来源于百度百科词条，其余内容由LearningYard学苑原创，如有侵权请沟通。

译文：

Hello, everyone. This is leanringyard college. Today is the second day of the new year. Xiaobian is here to wish you all the best in the new year.

In this festive moment, Xiaobian will not forget to bring you new knowledge, so today we continue to bring you a series of articles on Mathematica introductory tutorial. Todays theme is Shapley value method.

1. Executive summary

In the last issue, I combed the theories, unfamiliar terms and concepts used in the study on coordinated pricing of dual channel supply chain under the influence of discount rate of e-commerce platform (authors: Zhang Shen, Meng Qingchun, an Guozheng), as well as the ideas and methods of constructing the model in the paper. At the end, I mentioned the coordination scheme, which is based on Shapley value, And Xiaobian also knew this method for the first time, so he made a simple study a few days ago. In this issue, well take a look at what Shapley value is and how it is used in the article.

2. Shapley value method

Originally, I wanted to Baidu, but what Baidu said was a little unclear, so I went to HowNet to find relevant papers to study. This article describes the definition of Shapley value method in detail, and shows how to use it in supply chain coordination through a simple and specific numerical example.

Purpose: Based on my own understanding, there are multiple participants engaged in an economic activity. Under the optimal combination strategy, this activity will produce a maximum total benefit. They will adopt the optimal distribution scheme to maximize their benefits. This distribution scheme is the Shapley value method.

Definition: the above figure illustrates the definition of Shapley value method. First, the following conditions must be met: when there are no members, the income is zero; Overall rationality and individual rationality. Then, list the possible Federation sets. Finally, the income distributed by each member is calculated according to the formula.

I didnt explain too much about the parameters, so I referred to another article and explained each formula:

(1) : indicates that when there is no enterprise, the whole activity income is 0;

(2) : indicates super additivity, similar to 1 + 1 > = 2, indicating that the total income of enterprises in alliance is greater than or equal to the sum of their respective income when enterprises are not in alliance;

(3) : indicates the overall rationality. The sum of the income allocated by each member from the total income of the alliance is equal to the total income of the alliance;

(4) : indicates individual rationality. The income allocated by an individual from the alliance cannot be lower than the income completed alone.

(5) : represents the income of members. The income of player I given by Shapley is called Shapley value;

(6) : final weight calculation formula.

This is the whole Shapley value method model. The proof process will not be explained in the small series.

3. Practical application

After knowing the basic principle of Shapley value method, lets take a look at its specific application in this paper.

The supply chain in this paper only includes two decision makers, manufacturer and retailer. First, determine the income allocated by the manufacturer, and the alliance set I = {manufacturer, (manufacturer, retailer)}. First, determine the specific values of each parameter according to the income calculation formula, as shown in the table, and then calculate the results. The last row in the table in the figure below represents the income that the manufacturer can allocate under each alliance.

Adding the income distributed under the two alliances represents the manufacturers profit after coordination. Similarly, the profit of retailers can be calculated.

After determining the profit expression after coordination, make it equal to the profit expression during decentralized decision-making, and use the software to calculate the corresponding wholesale price.

END

This issue of sharing is over. We have learned all the theoretical knowledge, modeling ideas and methods of the article. In the next issue, we will start to use Mathematica for practical exercises. Lets look forward to it.

Those who are interested in Shapley value method can search for more materials and learn by themselves. You are also welcome to leave messages and communicate with Xiaobian!

Finally, I wish you a happy new year and new harvest in the New Year!