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# 经济代写|博弈论代考GAME THEORY代写|ECON90022 A congestion game

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## 经济代写|博弈论代考Game theory代写|A congestion game

Let $G=(V, E)$ be a finite graph, where we think of each directed edge $e=(v, w)$ as representing a road between cities $v$ and $w$. For each edge $e=(v, w)$ let $c_e: \mathbb{N} \rightarrow \mathbb{R}^{+}$be a monotone increasing congestion function, where $c_e(k)$ is the travel time when $k$ cars are on the road from $v$ to $w$.

Consider a finite set of players $N$. Each player $i$ has to travel from some city $v_i$ to some other city $w_i$, and so has to choose a path $s_i$ that connects $v_i$ to $w_i$. We assume that the chosen paths are always simple (i.e., do not repeat edges) and so we can think of $s_i$ simply as a subset of $E$.

Consider the game $G$ in which each player $i$ ‘s set of strategies is the set of all simple paths from $v_i$ to $w_i$. Given a strategy profile $s=\left(s_1, \ldots, s_n\right)$ and an edge $e$, we denote by $n_e(s)$ the number of players who travel on $e$ :
$$n_e(s)=\left|\left{i \in N: e \in s_i\right}\right| .$$

## 经济代写|博弈论代考Game theory代写|Potential games

Let $G=\left(N,\left{S_i\right},\left{u_i\right}\right)$ be a strategic form game. We say that $G$ is a potential game if there exists a $\Phi: S \rightarrow \mathbb{R}$ with the same property as in the example above: For every $s=\left(s_{-i}, s_i\right) \in S$ and $s^{\prime}=\left(s_{-i}, s_i^{\prime}\right) \in S_i$ it holds that
$$u_i\left(s^{\prime}\right)-u_i(s)=\Phi\left(s^{\prime}\right)-\Phi(s) .$$
The proof of 9.7 applies to any finite potential game, showing that they have no infinite better response paths. Thus better response dynamics always converges to a pure Nash equilibrium for finite potential games.

# 博弈论代写

## 经济代写|博弈论代考GAME THEORY代写|A CONGESTION GAME

$\mathrm{n}_{-} \mathrm{e}(\mathrm{s})=\backslash$ left $\mid$ left ${i \backslash$ in $\mathrm{N}: \mathrm{e}$ \in s_i $\backslash$ right $} \backslash$ right $\mid$.

## 经济代写|博弈论代考GAME THEORY代 写|POTENTIAL GAMES

$$u_i\left(s^{\prime}\right)-u_i(s)=\Phi\left(s^{\prime}\right)-\Phi(s) .$$
9.7 的证明适用于任何有限潜在博恋，表明它们没有无限好的响应路径。因此，更好的响应动态总是收敛到有限潜在 博栾的纯纳什均衡。

## 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。