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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|博弈论代写Game Theory代考|Congestion Games

Congestion means that a shared resource, such as a road, becomes more costly when more people use it. In a congestion game, multiple players decide on which resource to use, with the aim to minimize their cost. This interaction defines a game because the cost depends on what the other players do.

We present congestion networks as a model of traffic where individual users choose routes from an origin to a destination. Fach edge of the network is congested by creating a cost to each user that weakly increases with the number of users of that edge. The central result (Theorem 2.2) states that every congestion game has an equilibrium where no user can individually reduce her cost by changing her chosen route.

Like Chapter 1, this is another introductory chapter to an active and diverse area of game theory, where we can quickly show some important and nontrivial results. It also gives an introduction to the game-theoretic concepts of strategies and equilibrium before we develop the general theory.

Equilibrium is the result of selfish routing by the users who individually minimize their costs. This is typically not the socially optimal outcome, which could have a lower average cost. A simple example is the “Pigou network” (see Section 2.2), named after the English economist Arthur Pigou, who introduced the concept of an externality (such as congestion) to economics. Even more surprisingly, the famous Braess paradox, threated in Section 2.3, shows that adding capacity to a road network can worsen equilibrium congestion.

In Section $2.4$ we give the general definition of congestion networks. In Section $2.5$ we prove that every congestion game has an equilibrium. This is proved with a cleverly chosen potential function, which is a single function that simultaneously reflects the change in cost of every user when she changes her strategy. Its minimum over all strategy choices therefore defines an equilibrium.
In Section 2.6, we explain the wider context of the presented model. It is the discrete model of atomic (non-splittable) flow with finitely many individual users who choose their traffic routes. As illustrated by the considered examples, the resulting equilibria are often not unique when the “last” user can optimally choose between more than one edge. The limit of many users is the continuous model of splittable flow where users can be infinitesmally small fractions of a “mass” of users who want to travel from an origin to a destination. In this model the equilibrium is often unique, but its existence proof is more technical and not given here.
We also mention the Price of Anarchy that compares the worst equilibrium cost with the socially optimal cost. For further details we give references in Section 2.7.

## 经济代写|博弈论代写Game Theory代考|The Pigou Network

Getting around at rush hour takes much longer than at less busy times. Commuters take buses where people have to queue to board. There are more buses that fill the bus lanes, and cars are in the way and clog the streets because only a limited number can pass each green light. Commuters can take the bus, walk, or ride a bike, and cars can choose different routes. Congestion, which slows everything down, depends on how many people use certain forms of transport, and which route they choose. We assume that people try to choose an optimal route, but what is optimal depends on what others do. This is an interactive situation, which we call a game. We describe a mathematical model that makes the rules of this game precise.

Figure $2.1$ shows an example of a congestion network. This particular network has two nodes, an origin $o$ and a destination $d$, and two edges that both connect $o$ to $d$. Suppose there are several users of the network who all want to travel from $o$ to $d$ and can choose either edge. Each edge has a cost $c(x)$ associated with it, which is a function that describes how costly it is for each user to use that edge when it has a flow or load of $x$ users. The top edge has constant $\operatorname{cost} c(x)=2$, and the bottom edge has $\operatorname{cost} c(x)=x$. The cost could for example be travel time, or incorporate additional monetary costs. The cost is the same for all users of the edge. The cost for each user of the top edge, no matter how many people use it, is always 2, whereas the bottom edge has cost 1 for one user, 2 for two users, and so on (and zero cost for no users, but then there is no one to profit from taking that zero-cost route).

Suppose the network in Figure $2.1$ is used by two users. They can either both use the top edge, or choose a different edge each, or both use the bottom edge. If both use the top edge, both pay cost 2, but if one of them switches to the bottom edge, that user will travel more cheaply and only pay cost 1 on that edge. We say that this situation is not in equilibrium because at least one user can improve her cost by changing her action. Note that we only consider the unilateral deviation of a single user in this scenario. If both would simultaneously switch to the bottom edge, then this edge would be congested with $x=2$ and again incur cost 2 , which is no improvement to the situation that both use the top edge.

# 博弈论代考

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师