Posted on Categories:Game theory , 博弈论, 经济代写

# 经济代写|博弈论代考Game theory代写|ECO467 Correlated equilibria

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## 经济代写|博弈论代考Game theory代写|Motivating example

Aumann [4] introduced the notion of a correlated equilibrium. Consider the following game, which is usually called “Chicken”. There are two drivers who arrive at the same time to an intersection. Each one would like to drive on (strategy D) rather than yield (strategy Y), but if both drive then they run the risk of damaging the cars. If both yield time is wasted, but no egos are hurt. The payoff matrix is the following.

This game has three Nash equilibria: two pure $((Y, D)$ and $(D, Y))$ and one mixed, in which each player drives with probability $1 / 3$.
Exercise 3.27. Show that this is indeed a mixed equilibrium.
The players’ expected utilities in these equilibria are $(0,5),(5,0)$ and $(2,2)$.
A natural way to resolve this conflict is by the installation of a traffic light which would instruct each player whether to yield or drive. For example, the light could choose uniformly at random from $(Y, D)$ and $(D, Y)$. It is easy to convince oneself that a player has no incentive to disobey the traffic light, assuming that the other player is obeying it. The players’ utilities in this case become $(2.5,2.5)$.

One could imagine a traffic light that chooses from ${(Y, Y),(Y, D),(D, Y)}$, where the first option is chosen with probability $p$ and the second and third are each chosen with probability $(1-p) / 2$. Now, given that a player is instructed to drive, she knows that the other player has been instructed to yield, and so, if we again assume that the other player is obedient, she has no reason to yield.

Given that a player has been instructed to yield, she knows that the other player has been told to yield with conditional probability $p_Y=p /(p+(1-p) / 2)$ and to drive with conditional probability $p_D=((1-p) / 2) /(p+(1-p) / 2)$. Therefore, her utility for yielding is $3 p_Y$, while her utility for driving is $5 p_Y-4 p_D$. It thus follows that she is not better off disobeying, as long as $3 p_Y \geq 5 p_Y-4 p_D$. A simple calculation shows that this is satisfied as long as $p \leq 1 / 2$

Now, each player’s expected utility is $3 p+5(1-p) / 2$. Therefore, if we choose $p=1 / 2$, the players’ expected utilities are $(2.75,2.75)$. In this equilibrium the sum of the players’ expected utilities is larger than in any Nash equilibrium.

## 经济代写|博弈论代考Game theory代写|Definition

We now generalize and formalize this idea. Let $G=\left(N,\left{S_i\right}_{i \in N},\left{u_i\right}_{i \in N}\right)$ be a finite game. A distribution $\mu \in \Delta S$ is a correlated equilibrium if for every player $i$ and every $s_i, t_i \in S_i$ it holds that
$$\sum_{s_{-i}} \mu\left(s_{-i}, s_i\right) u_i\left(s_{-i}, s_i\right) \geq \sum_{s_{-i}} \mu\left(s_{-i}, s_i\right) u_i\left(s_{-i}, t_i\right) .$$
Player $i$ ‘s expected utility under a correlated equilibrium $\mu$ is simply $\mathbb{E}_{s \sim \mu}\left[u_i(s)\right]$.
Note that for given $i$ and $s_i$, the condition in (3.2) is closed (i.e., if each of a converging sequence $\left{\mu_n\right}$ of distributions satisfies it then so does $\lim _n \mu_n$ ). Note also that if $\mu_1$ and $\mu_2$ satisfy (3.2) then so does any convex combination of $\mu_1$ and $\mu_2$. These observations immediately imply the following claim.

# 博弈论代写

## 经济代写|博亦论代考Game theory代写|Motivating example

Aumann [4]引入了相关均衡的概念。考虑以下游戏，通常称为“Chicken”。有两个司机同时到达一个十字路口。每个人都楒继 续开车（策略 D) 而不是屈服（策略 Y)，但如果两人都开车，那么他们就有损坏汽车的风险。如果双方都浪费了时间，但没有伤 害到自尔。收益矩轵如下。

$p_D=((1-p) / 2) /(p+(1-p) / 2)$. 因此，她屈服的效用是 $3 p_Y$ ，而她的驾驶效用是 $5 p_Y-4 p_D$. 由此可见，她不服火并不会 更好，只要 $3 p_Y \geq 5 p_Y-4 p_D$. 一简单的计算表明，只要满足 $p \leq 1 / 2$

## 经济代写|博弈论代考Game theory代写|Definition

$$\sum_{s_{-i}} \mu\left(s_{-i}, s_i\right) u_i\left(s_{-i}, s_i\right) \geq \sum_{s_{-i}} \mu\left(s_{-i}, s_i\right) u_i\left(s_{-i}, t_i\right) .$$

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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。