分享知识、传播快乐、增长见闻、留下美好，大家好，这里是Leanringyard学院。今天小编将继续为大家带来Mathematica入门教程系列文章，今天的主题就是：Mathematica在Stackelberg博弈中具体运用。
Share knowledge, spread happiness, increase knowledge and leave beauty. Hello, everyone, this is leanringyard college. Today, I will continue to bring you a series of articles on Mathematica introductory tutorial. Today’s theme is: the specific application of Mathematica in Stackelberg game.
内容提要（Executive summary）
前面我们已经学习了一些常用的Mathematica软件的操作，这些知识都是在后面能帮助我们进行完成运算的基础知识。
（1）通读全文，掌握主要研究内容；细读，梳理模型的构建的步骤；
（2）掌握文中使用的理论，陌生名词，概念等；论文构建模型的思路与方法；
（3）提炼模型，使用Mathematica进行模型构建，进行仿真分析；
Previously, we have learned the operation of some commonly used Mathematica software. These knowledge are the basic knowledge that can help us complete the operation later.
After consulting teachers, the editor will study Mathematica software with a paper in the Journal of system management – Research 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). For the study of this article, I divide it into the following parts:
(1) Read through the full text and master the main research contents; Read carefully and sort out the steps of model construction;
(2) Master the theories, unfamiliar nouns and concepts used in the text; The ideas and methods of constructing the model;
(3) Refine the model, use Mathematica to build the model and carry out simulation analysis;
论文介绍（Paper introduction）
研究背景：（1）电商平台扣点率的上涨对企业入驻平台的动机、产品售价产生显著影响，故研究电商平台扣点率影响下双渠道供应链定价问题具有重要现实意义；（2）电子商务的出现，重塑了消费者的消费习惯，大型制造企业开拓电商渠道，由传统销售渠道向双渠道供应链转变。
研究对象：电商平台扣点率影响下的双渠道供应链。
研究内容：构建了追求供应链整体利润最大化的集中式定价模型，得出最优线上、线下渠道销售价格，分析了扣点率影响下的价格、销量与利润变化，逐步确定了扣点率的决策范围。然后以分散式决策下企业最优利润，为基准设计基于 Shapley值的协调定价方案以确定最优线下渠道批发价格，给出了协调后的最优定价。
研究方法：Stackelberg博弈；Shapley值法。
Research background: (1) the rise of discount rate of e-commerce platform has a significant impact on the motivation of enterprises to settle in the platform and product price. Therefore, it is of great practical significance to study the pricing of dual channel supply chain under the influence of discount rate of e-commerce platform; (2) The emergence of e-commerce has reshaped consumers’ consumption habits. Large manufacturing enterprises have developed e-commerce channels and changed from traditional sales channels to dual channel supply chain.
Research object: Dual Channel Supply Chain under the influence of discount rate of e-commerce platform.
Research content: This paper constructs a centralized pricing model to maximize the overall profit of the supply chain, obtains the optimal online and offline channel sales price, analyzes the changes of price, sales volume and profit under the influence of the discount rate, and gradually determines the decision-making scope of the discount rate. Then, based on the optimal profit of enterprises under decentralized decision-making, a coordinated pricing scheme based on Shapley value is designed to determine the optimal offline channel wholesale price, and the coordinated optimal pricing is given.
Research methods: Stackelberg game; Shapley value method.
思路梳理（Train of thought combing）
根据文章自身双渠道供应链模型构建图，得到模型中各参数变量，以及整个交易流程。
1.首先，设定线上线下各渠道的产品需求，基于需求和产品价格得到制造商，零售商，线上渠道，线下渠道的利润表达式，以及整体供应链利润。
2.先构建供应链集中决策下的供应链决策模型；通过利润函数求出各渠道销售的最优价格，在进行命题分析。
3.构建分散决策下双渠道供应链定价模型，采用逆向归纳算法计算各渠道的最优批发价格，零售价格，并将这些解代入进行利润表达式种，得到最优利润。
4.在2.3步基础上，按照Shapley值得规则确定协调之后的利润表达式，再通过求解得到最优批发价格。
According to the construction diagram of the dual channel supply chain model, the parameter variables in the model and the whole transaction process are obtained.
1. Firstly, set the product demand of online and offline channels, and obtain the profit expression of manufacturers, retailers, online channels and offline channels and the overall supply chain profit based on the demand and product price.
2. Firstly, build the supply chain decision-making model under the centralized supply chain decision-making; Through the profit function, the optimal sales price of each channel is obtained, and the proposition analysis is carried out.
3. Build the pricing model of dual channel supply chain under decentralized decision-making, use the reverse induction algorithm to calculate the optimal wholesale price and retail price of each channel, and substitute these solutions into the profit expression to obtain the optimal profit.
4. On the basis of step 2.3, the profit expression after coordination is determined according to Shapley value rule, and then the optimal wholesale price is obtained by solving.
涉及函数（Involving function）
在整个模型构建以及求解过程中，可能用到的知识点：
- 赋值“=”
- D[]:求偏导函数
- Solve[]:解方程函数
- 函数定义f[x_ ]
- plot[]:绘制函数图像
- “/.”:将参数值代入原表达式
- Fullsimplify[]:化简复杂的表达式
Possible knowledge points in the whole process of model construction and solution:
Assignment “=”
D []: find partial derivative function
Solve []: solve equation function
Function definition f[x#]
Plot []: draw function image
“/.”: Substitute the parameter value into the original expression
Fullsimplify []: simplify complex expressions
今天的分享就到这里了。如果您对今天的文章有独特的想法，欢迎给我们留言，让我们相约明天，祝您今天过得开心快乐！That’s it for today’s sharing.If you have unique ideas about today’s article, please leave us a message,Let us meet tomorrow,I wish you a happy day today!
参考资料：
[1]谷歌翻译、百度百科，Mathematica软件；
[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.
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