## 经济代写|博弈论代写Game Theory代考|ECOS3012

statistics-lab™ 为您的留学生涯保驾护航 在代写博弈论Game Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写博弈论Game Theory代写方面经验极为丰富，各种代写博弈论Game Theory相关的作业也就用不着说。

• 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|STRATEGIC GAMES

Prisoner’s Dilemma is a canonical example of a strategic game because, as we will see shortly, it typifies many scenarios that confront decision makers. Further, being a simple scenario, it can be used to illustrate many of the fundamental concepts of game theory, and it also clearly demonstrates a fundamental dilemma in our (human) decision-making processes.

We model this scenario as a strategic game in which the two suspects, each confined in a separate interrogation room, are the players. We will often refer to our two players in strategic games as Rose and Colin. (This convention helps later to emphasize the distinction between row and column players and was popularized by Phil Straffin in his book Game Theory and Strategy [110].) They each have two strategies available to them which we name Quiet and Confess. Table $3.1$ lists each of the strategy profiles in the form (Rose, Colin) and the resulting outcome.

We assume that each suspect is primarily concerned about their own sentence and wants to minimize it. Table $3.2$ provides payoffs (a common synonym for utilities) for each player. Here we use the utility function 6 minus the number of years in prison; this is consistent with the player preferences. Based on our assumptions, these payoffs are ordinal. For these payoffs to also be vNM, we would need to assume that the suspects are risk neutral in the number of years to be served in prison.

Tables $3.1$ and $3.2$ complete the construction of the model by identifying the strategies, outcomes, and payoffs. We will refer to this model of the Prisoner’s Dilemma scenario as the Prisoner’s Dilemma strategic game.

We are now ready to look for a solution that maximizes the payoffs to the players. By observing that $5>3$, we see that Confess is the best response strategy for Rose if she knows that Colin will choose Quiet. Further, we can observe that Confess is also a best response for Rose if she knows Colin will choose Confess. We formalize the definition of a best response strategy below.

## 经济代写|博弈论代写Game Theory代考|FINALJEOPARDY

As we observed in Section 3.1, the phrase “Prisoner’s Dilemma” has been used to describe many real-world scenarios; however, not all of these scenarios actually fit the mathematical definition. This can only be revealed by constructing and analyzing a model of the scenario.

We examine a situation in Jeopardy! which fans have identified as a Prisoner’s Dilemma. In the Final Jeopardy round, each contestant makes a wager as to whether they can answer a specific question correctly. When making the wager, contestants know the category of the question, but not the question itself, and the amount of money each of the other contestants has available. Each player’s wager can be between 0 and their current winnings. Depending on whether the contestant answers the question correctly, they win or lose the amount of money wagered. The contestant with the most money after this final round of play wins the game. The winner keeps all of their winnings, and the other two contestants lose essentially all of their money. If there is a tie at the end of the round, a simple, essentially random, tie-breaker rule is applied to identify the winner.

The so-called Prisoner’s Dilemma situation occurs when two contestants are tied for the lead, and the third contestant has less than half of the money of either of the first two contestants. For simplicity we will assume that it is contestants 1 and 2 who are tied with the most money.

In this situation, aficionados of Jeopardy! often refer to “Jeek’s Rule,” which asserts that while they could wager any amount up to their current winnings, contestants 1 and 2 should either wager nothing or everything. We discuss the reasonableness of this rule and then make it an assumption when we define our strategic game.

Let $E$ ‘ be the amount of money contestants 1 and ‘ 2 have each won at the time Final Jeopardy begins. Let $w_i$ denote the wager of contestant $i$ and suppose that contestant 1 ‘s wager satisfies $0<w_1<E$. There are four cases to consider:

Case 1: Both contestants answer the question correctly. In this case, if $w_1<w_2$, contestant 1 regrets not wagering $E$ in order to win. If $w_1 \geq w_2$, then contestant 1 regrets not wagering $E$ to maximize their winnings.

Case 2: Contestant 1 answers the question correctly and contestant 2 does not. Here contestant 1 regrets not wagering $E$ in order to maximize their winnings.

Case 3 : Contestant 1 answers the question incorrectly and contestant 2 answers correctly. Then contestant 1 is indifferent about their bet unless $w_2=0$, in which case they regret not wagering $w_1=0$.

Case 4: Both contestants answer the question incorrectly. Here, if $w_1 \geq w_2$, contestant 1 regrets not wagering $w_1=0$ in order to win. If $w_1<w_2$, then contestant 1 regrets not wagering $w_1=0$ to maximize their winnings.

# 博弈论代考

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 经济代写|博弈论代写Game Theory代考|ECON6025

statistics-lab™ 为您的留学生涯保驾护航 在代写博弈论Game Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写博弈论Game Theory代写方面经验极为丰富，各种代写博弈论Game Theory相关的作业也就用不着说。

• 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|CONSTRUCTING UTILITIES

We have already suggested how we might construct a utility function to model a player’s choices when there are a finite number of outcomes. Ordinal preferences can be revealed by asking the player to choose among all outcomes and assign those outcomes the highest utility, asking the player to choose among all outcomes not previously chosen and assign those outcomes the second highest utility, and so forth. vNM preferences can be obtained by asking the player to name the highest and lowest ranked outcomes $o_h$ and $o_l$, assign utilities of $u\left(o_h\right)=1$ and $u\left(o_l\right)=0$ to these outcomes, and then for each remaining outcome $o$ determine a probability $p$ for which the player would be willing to choose either the outcome $o$ or the lottery $(1-p) o_l+p o_h$ and assign $u(o)=p$.

In this section, we examine four specific scenarios to illustrate a variety of ways utility functions may be created.

To model Self-Interest and Other-Interest, we simplify our scenario to examine the monthly salaries of the job offers for each spouse. Suppose Scarlett and Regis receive $\$ x$thousand and$\$y$ thousand, respectively; we will denote this by $(x, y)$. Consider the following four possible outcomes: $(7,0),(6,6),(5,7)$, and $(1,6)$. If Scarlett is exclusively self-interested, she would rank order these outcomes in the given order. If Scarlett is primarily interested in Regis receiving money and only secondarily interested in receiving money for herself, Scarlett would rank order the outcomes $(5,7),(6,6),(1,6)$, and $(7,0)$. If Scarlett had a mixture of self-interest, other-interest, and a desire for equity, she might rank order the outcomes $(6,6),(5,7),(7,0)$, and $(1,6)$.

In fact, this last rank order would be obtained if Scarlett considered $\$ 1,000$given to Regis to be worth the same to her as her receiving$\$500$, suggesting the utility function $u(x, y)=x+0.5 y$. Of course, this is only an ordinal utility function unless, at minimum, Scarlett is indifferent between the outcome $(7,0)$ with utility $u(7,0)=7$ and the lottery $L=0.6(6,6)+0.4(1,6)$ with utility
$$u(L)=0.6 u(6,6)+0.4 u(1,6)=0.6(9)+0.4(4)=7 .$$
This example demonstrates how we can incorporate both self-interest and altruistic interests into a player’s utility function. Therefore, maximizing a utility function does not necessarily imply selfishness, but rather achieving the most preferred outcome based on the player’s interests.

## 经济代写|博弈论代写Game Theory代考|DETERMINING RISK

In the duopoly scenario, and in most other economic models, the utility of an outcome is cquivalent to some dollar value associated with the outcome. While we can see how dollar values might capture the intensity of a player’s preferences, dollar values are not necessarily vNM utilities. For example, receiving $\$ 11.00$instead of$\$10.00$ means significantly more than receiving $\$ 1001.00$instead of$\$1000.00$ to most people. To explore this difference, we consider the relationship between the expected utility of a lottery, as given by the Expected Utility Hypothesis, and the utility of the expected value of the lottery.

Consider the following raffle: For $\$ 25$, you can purchase a$\frac{1}{400}$chance for a$\$10,000$ college scholarship. We can represent this lottery with our usual notation
$$\left.\frac{399}{400}(\text { losing } \ 25)+\frac{1}{400} \text { (winning } \ 9,975\right) \text {, }$$ but since the outcomes are numerical, we can calculate the expected monetary value of the raffle as
$$\frac{399}{400}(-\ 25)+\frac{1}{400}(\ 9,975)=\ 0 .$$
The expected monetary value of entering or not entering the raffle is the same, however, entering the raffle involves a small chance of a large gain offset by a large chance of a small loss, while not entering the raffle involves no chance of a gain or a loss. Entering the raffle involves risk while not entering the raffle does not.

Most parents of college students are willing to enter the raffle, but many college students themselves are not. For the college parents,
$$\left.\left.u\left(\frac{399}{400} \text { (losing } \ 25\right)+\frac{1}{400} \text { (winning } \ 9,975\right)\right)>u(\ 0),$$
but for the students themselves,
$$\left.u\left(\frac{399}{400}(\text { losing } \ 25)+\frac{1}{400} \text { (winning } \ 9,975\right)\right)<u(\ 0) .$$
For the parents, the utility of the lottery is greater than the utility of the expected value, making them risk loving in this scenario. On the other hand, the students are risk adverse since the utility of the lottery is less than the utility of the expected value. This principle holds in general, as we describe in the following definition.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|CONSTRUCTING UTILITIES

$$u(L)=0.6 u(6,6)+0.4 u(1,6)=0.6(9)+0.4(4)=7 .$$

## 经济代写|博弈论代写Game Theory代考|DETERMINING RISK

$$\left.\left.\frac{399}{400} \text { ( losing } \ 25\right)+\frac{1}{400} \text { (winning } \ 9,975\right) \text {, }$$

$$\frac{399}{400}(-\ 25)+\frac{1}{400}(\ 9,975)=\ 0 .$$

$$u\left(\frac{399}{400}(\text { losing } \ 25)+\frac{1}{400}(\text { winning } \ 9,975)\right)>u(\ 0)$$

$$u\left(\frac{399}{400}(\text { losing } \ 25)+\frac{1}{400}(\text { winning } \ 9,975)\right)<u(\ 0) .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 经济代写|博弈论代写Game Theory代考|ECON2112

statistics-lab™ 为您的留学生涯保驾护航 在代写博弈论Game Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写博弈论Game Theory代写方面经验极为丰富，各种代写博弈论Game Theory相关的作业也就用不着说。

• 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|ORDINAL UTILITIES

In this scenario, there are four possible outcomes: Scarlett leaves with vanilla ice cream, which we will label $V$, leaves with chocolate ice cream, labeled $C$, leaves with strawberry ice cream, labeled $S$, or leaves with no ice cream, labeled $N$. Our outcome set is then ${V, C, S, N}$. The rules dictate that Scarlett may choose any outcome that is available.
To model Scarlett’s preferences and build a utility function, we consider the choices she would make when certain outcomes are available. Supposing that all potential outcomes are available, Scarlett would choose $V$; if $C$ were not an available outcome, Scarlett would choose $S$; if only $V$ and $N$ were available outcomes, Scarlett would flip a coin to determine whether to select $V$ or $N$; and if only $C$ and $S$ were available outcomes, Scarlett would be unable to make a choice. While this behavior would be possible, most would find it bizarre: Scarlett’s choice of $V$ when presented with the outcome set ${V, C, S, N}$ suggests she prefers $V$ to $S$, but Scarlett’s choice of $S$ when presented the outcome subset ${V, S, N}$ suggests she prefers $S$ to $V$. To avoid such absurd possibilities, we will assume that individual behavior is governed by self-consistent internal preferences over the outcomes, which is reflected in the mathematical definition of ordinal preferences below.

Definition 2.1.1. A player $i$ is said to have ordinal preferences among outcomes if there exists a utility function $u_i$ from the set $O$ of outcomes into the real numbers, $\mathbb{R}$, such that whenever presented with a subset $O^{\prime} \subseteq O$ of outcomes, player $i$ chooses any of the outcomes that maximize $u_i$ over all outcomes $o \in O^{\prime}$.

To ensure that ordinal preferences align with the players real-world choice behavior, we note that whenever player $i$ prefers outcome $o_j$ over outcome $o_k$, we should have $u_i\left(o_j\right)>$ $u_i\left(o_k\right)$, and when player $i$ is indifferent between $o_j$ and $o_k$ we should have $u_i\left(o_j\right)=u_i\left(o_k\right)$. We now ask under what conditions Scarlett’s outcome choice behavior can be modeled by ordinal preferences and describe three reasonable properties for a self-consistent set of choices. Since it is usually easier for a player to choose between two rather than among many outcomes, these properties will focus on pairwise choices.

First, we want Scarlett’s pairwise choices to be complete, meaning that whenever she is presented with a pair of outcomes, Scarlett is able to make a choice. Equivalently, for each pair of outcomes, $A$ and $B$, exactly one of the following conditions holds: (a) Scarlett chooses $A$ over $B$, (b) Scarlett chooses $B$ over $A$, or (c) Scarlett is willing to flip a coin to determine which outcome to choose (in this case we will often say Scarlett chooses either $A$ or $B$ ). Hence, this condition excludes the following option as a possibility: (d) Scarlett chooses neither $A$ nor $B$. (When we want the rules to allow a player to choose none of the options, we must include that as an outcome, as we did in the Ice Cream Parlor scenario.) For (a), we will assign utilities so that $u(A)>u(B)$. Likewise for (b) we will assign utilities so that $u(B)>u(A)$. Finally for (c), since Scarlett is willing to flip a coin to determine the outcome, we assume she is indifferent between $A$ and $B$ and assign $u(A)=u(B)$.

## 经济代写|博弈论代写Game Theory代考|VON NEUMANN-MORGENSTERN UTILITIES

A significant limitation of ordinal preferences and their associated utility functions is that they cannot describe the intensity of a player’s preference for a particular outcome. That is, they cannot capture the difference between Scarlett preferring vanilla ice cream over chocolate ice cream and Scarlett so strongly preferring vanilla that she would pay for it rather than have a free serving of chocolate. Notice how we have once again translated our intuitive sense of internal preference intensity into something that is observable (a real-world choice) so we can create a utility function based on these choices. While asking players to choose among outcomes that include the receipt or payment of money would be one observable way to determine intensity of preferences, we will take an approach that does not rely on the availability of money.

We begin by introducing a new, probability-based outcome called a lottery. Suppose that when a second customer, Regis, enters the ice cream parlor, he encounters a college student conducting a taste test involving different flavors of ice cream. The college student offers Regis the choice of either a sample of chocolate ice cream (his second-most favorite) or an unknown sample that is either vanilla (his favorite) or strawberry (his least favorite). The second option in this example is a simple lottery.

Definition 2.2.1. Given a set of outcomes, $O$, a simple lottery is a probability distribution over this set. When $O=\left{o_1, o_2, \ldots, o_m\right}$, a finite set, a simple lottery can be denoted by $p_1 o_1+p_2 o_2+\ldots+p_m o_m$ where $p_i$ is the probability of outcome $o_i$. A compound lottery is a probability distribution over other lotteries. Because an outcome $o \in O$ can be written as the simple lottery $1 o$, we see that $O \subset \mathcal{L}$, the set of all (simple and compound) lotteries.
Tó reveál thè strength of a player’s préference for one outcomé over another, wè must examine not nnly chnires hetween single nutromes hut chnices hetween single nutcomes and lotteries. Suppose Regis prefers vanilla $V$ over chocolate $C$ and prefers $C$ over strawberry $S$. The choice Regis makes between $C$ and the lottery $0.5 S+0.5 V$ tells us about the strength of his preference for $V$ over $C$ and for $C$ over $S$. If he would choose either of the two possibilities, then the strength of Regis’s preference for $C$ is exactly halfway between $S$ and $V$. If Regis were to choose $C$ over $0.5 S+0.5 \mathrm{~V}$, it reveals that his preference intensity for $C$ is closer to $V$ than to $S$. If Regis were willing to choose either $C$ or the lottery $0.1 S+0.9 \mathrm{~V}$, it would reveal that his preference for $V$ over $C$ is very small and his preference for $C$ over $S$ is relatively large. However, if he were instead willing to choose $C$ or the lottery $0.9 S+0.1 \mathrm{~V}$, it would reveal a strong preference for $V$ over $C$ and that his preferences for $C$ over $S$ is small. When a player is willing to choose either of two lotteries, this reveals the player is indifferent between these choices. This motivates the following generalization of the utility function concept.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|VON NEUMANN-MORGENSTERN UTILITIES

O=\left{0_1，o_2, Vdots, o_m\right } } \text { ，一个有限集，一个简单的彩票可以表示为 }
$p_1 o_1+p_2 o_2+\ldots+p_m o_m$ 在哪里 $p_i$ 是结果的概率 $o_i$. 复合彩票是其他彩票的概率分布。因为一个结 果 $o \in O$ 可以写成简单的彩票 $l o$ ，我们看到 $O \subset \mathcal{L}$ ，所有（简单和复合） 彩票的集合。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 经济代写|博弈论代写Game Theory代考|ECON3050

statistics-lab™ 为您的留学生涯保驾护航 在代写博弈论Game Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写博弈论Game Theory代写方面经验极为丰富，各种代写博弈论Game Theory相关的作业也就用不着说。

• 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|Backward induction, Kuhn’s Theorem

Let $G=\left(N, A, H, O, o, P,\left{\leq_i\right}_{i \in N}\right)$ be an extensive form game with perfect information. Recall that $A(\varnothing)$ is the set of allowed initial actions for player $i=P(\varnothing)$. For each $b \in A(\varnothing)$, let $s^{G(b)}$ be some strategy profile for the subgame $G(b)$. Given some $a \in A(\varnothing)$, we denote by $s^a$ the strategy profile for $G$ in which player $i=P(\varnothing)$ chooses the initial action $a$, and for each action $b \in A(\varnothing)$ the subgame $G(b)$ is played according to $s^{G(b)}$. That is, $s_i^a(\varnothing)=a$ and for every player $j, b \in A(\varnothing)$ and $b h \in H \backslash Z, s_j^a(b h)=s_j^{G(b)}(h)$.
Lemma 2.11 (Backward Induction). Let $G=\left(N, A, H, O, o, P,\left{\leq_i\right}_{i \in N}\right)$ be a finite extensive form game with perfect information. Assume that for each $b \in A(\varnothing)$ the subgame $G(b)$ has a subgame perfect equilibrium $s^{G(b)}$. Let $i=P(\varnothing)$ and let a be the $>_i$-maximizer over $A(\varnothing)$ of $o_a\left(s^{G(a)}\right)$. Then $s^a$ is a subgame perfect equilibrium of $G$.

Proof. By the one deviation principle, we only need to check that $s^a$ does not have deviations that differ at a single history. So let $s$ differ from $s^a$ at a single history $h$.

If $h$ is the empty history then $s=s^{G(b)}$ for $b=s_i(\varnothing)$. In this case $o\left(s^a\right)>_i o(s)=o_b\left(s^{G(b)}\right)$, by the definition of $a$ as the maximizer of $o_a\left(s^{G(a)}\right)$.

Otherwise, $h$ is equal to $b h^{\prime}$ for some $b \in A(\varnothing)$ and $h^{\prime} \in H_b$, and $o(s)=o_b(s)$. But since $s^a$ is a subgame perfect equilibrium when restricted to $G(b)$ there are no profitable deviations, and the proof is complete.
Kuhn [22] proved the following theorem.
Theorem $2.12$ (Kuhn, 1953). Every finite extensive form game with perfect information has a subgame perfect equilibrium.

Given a game $G$ with allowed histories $H$, denote by $\ell(G)$ the maximal length of any history in $H$.

Proof of Theorem 2.12. We prove the claim by induction on $\ell(G)$. For $\ell(G)=0$ the claim is immediate, since the trivial strategy profile is an equilibrium, and there are no proper subgames. Assume we have proved the claim for all games $G$ with $\ell(G)<n$.

Let $\ell(G)=n$, and denote $i=P(\varnothing)$. For each $b \in A(\varnothing)$, let $s^{G(b)}$ be some subgame perfect equilibrium of $G(b)$. These exist by our inductive assumption, as $\ell(G(b))<n$.

Let $a^* \in A(\varnothing)$ be a $\leq_i$-maximizer of $o\left(s^{a^}\right)$. Then by the Backward Induction Lemma $s^{a^}$ is a subgame perfect equilibrium of $G$, and our proof is concluded.

## 经济代写|博弈论代写Game Theory代考|Classical examples

• Extensive form game with perfect information. Let $G=\left(N, A, H, P,\left{u_i\right}_{i \in N}\right)$ be an extensive form game with perfect information, where, instead of the usual outcomes and preferences, each player has a utility function $u_i: Z \rightarrow \mathbb{R}$ that assigns her a utility at each terminal node. Let $G^{\prime}$ be the strategic form game given by $G^{\prime}=\left(N^{\prime},\left{S_i\right}_{i \in N},\left{u_i\right}_{i \in \mathbb{N}}\right)$, where
$-N^{\prime}=N$.
• $S_i$ is the set of $G$-strategies of player $i$.
• For every $s \in S, u_i(s)$ is the utility player $i$ gets in $G$ at the terminal node at which the game arrive when players play the strategy profile $s$.

We have thus done nothing more than having written the same game in a different form. Note, however, that not every game in strategic form can be written as an extensive form game with perfect information.

Exercise 3.1. Show that $s \in S$ is a Nash equilibrium of $G$ iff it is a Nash equilibrium of $G^{\prime}$.

Note that a disadvantage of the strategic form is that there is no natural way to define subgames or subgame perfect equilibria.

• Matching pennies. In this game, and in the next few, there will be two players: a row player (R) and a column player (C). We will represent the game as a payoff matrix, showing for each strategy profile $s=\left(s_R, s_C\right)$ the payoffs $u_R(s), u_C(s)$ of the row player and the column player.

# 博弈论代考

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。