## 经济代写|博弈论代写Game Theory代考|ECON3503

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|Further Discussions

Game theory attempts to bring mathematical precision to decision-making so that best strategies can be played, even in the arms race between the learner and the adversary. In game theory, we assume the players are rational-they seek to maximize their payoffs or minimize their losses. This assumption is not necessarily valid in real life, especially in cybersecurity domains. What appears irrational to one player may be rational to the opponent player. For example, when we model a game between airport security and terrorists, it may not be a good idea to view the terrorists based on our own experiences. When the opponent is not rational or simply plays poorly, it is important to realize that continuing to play the equilibrium strategy will lead to a losing situation or the loss of opportunity to exploit the opponent’s weaknesses.

Behavioral economists have long questioned the strict assumptions about rationality in existing theories in game theory (Aumann 1997). The actual decision-making by individuals is often irrational even in simple decision problems. Individuals typically fail to select the best response from a set of choice. As a matter of fact, optimization is so difficult that individuals are often unable to come up with their best responses to their opponents’ moves, unlike what the rational decision theory has always assumed. In response, theories that relax the rationality constraints have been proposed among which include: Quantal Response Equilibrium (QRE) (McKelvey and Palfrey 1995), a solution concept that promotes an equilibrium notion with bounded rationality that acknowledges the possibility that players do not always play a best response; Prospect Theory (Kahneman and Tversky 1979; Tversky and Kahneman 1992), a solution that introduces payoffs with respect to a reference point at which there is larger slope for losses than for gains and agents tend to overestimate small probabilities and underestimate large probabilities.

The same debate is applicable to mixed strategy games. In the airport security game, how the airport deploys security patrols depends on the response from the terrorists. If the rationality of the terrorists is predictable, it would be best for the airport to deploy its security patrols randomly by playing a mixed strategy. However, when there is a good reason to believe that the terrorists are not going to play the equilibrium strategy, airport security may be better off by playing pure strategies. Generally speaking, unless the odds are strongly in our favor, playing equilibrium strategies would be our best choice. Nevertheless, we should always keep in mind that our calculation of rationality may lead to different behavior, new rules may need to be defined in the game to take this into consideration.

While there is a growing interest in applying machine learning to different data domains and deploying machine learning algorithms in real systems, it has become imperative to understand vulnerabilities of machine learning in the presence of adversaries. To that end, adversarial machine learning (Kurakin et al. 2016a; Vorobeychik and Kantarcioglu 2018; Shi et al. 2018b) has emerged as a critical field to enable safe adoption of machine learning subject to adversarial effects. One example that has attracted recent attention involves machine learning applications offered to public or paid subscribers via APIs; e.g. Google Cloud Vision (2020) provides cloud-based machine learning tools to build machine learning models. This online service paradigm creates security concerns of adversarial inputs to different machine learning algorithms ranging from computer vision to NLP (Shi et al. 2018c,d). As another application domain, automatic speech recognition and voice controllable systems were studied in terms of the vulnerabilities of their underlying machine learning algorithms (Vaidya et al. 2016; Zhang et al. 2017). As an effort to identify vulnerabilities in autonomous driving, attacks on self-driving vehicles were demonstrated in Kurakin et al. (2016), where the adversary manipulated traffic signs to confuse the learning model.

The manipulation in adversarial machine learning may happen during the training or inference (test) time, or both. During the training time, the goal of the adversary is to provide wrong inputs (features and/or labels) to the training data such that the machine learning algorithm is not properly trained. During the test time, the goal of the adversary is to provide wrong inputs (features) to the machine algorithm such that it returns wrong outputs. As illustrated in Figure 14.1, attacks built upon adversarial machine learning can be categorized as follows.

1. Attack during the test time.
a. Inference (exploratory) attack: The adversary aims to infer the machine learning architecture of the target system to build a shadow or surrogate model that has the same functionality as the original machine learning architecture (Barreno et al. 2006; Tramer et al. 2016; Wu et al. 2016; Papernot et al. 2017; Shi et al. 2017; Shi et al. 2018b). This corresponds to a white-box or black-box attack depending on whether the machine learning model such as the deep neural network structure is available to the adversary, or not. For a black-box attack, the adversary queries the target classifier with a number of samples and records the labels. Then, it uses this labeled data as its own training data to train a functionally equivalent (i.e. statistically similar) deep learning classifier, namely a surrogate model. Once the machine learning functionality is learned, the adversary can use the inference results obtained from the surrogate model for subsequent attacks such as confidence reduction or targeted misclassification.
b. Membership inference attack: The adversary aims to determine if a given data sample is a member of the training data, i.e. if a given data sample has been used to train the machine learning algorithm of interest (Nasr et al. 2018; Song et al. 2018; Jia et al. 2019; Leino and Fredrikson 2020). Membership inference attack is based on the analysis of overfitting to check whether a machine learning algorithm is trained for a particular data type, e.g. a particular type of images. By knowing which type of data the machine learning algorithm is trained to classify, the adversary can then design a subsequent attack more successfully.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|Further Discussions

1. 在测试时间内攻击。
A。推理（探索性）攻击：对手旨在推断目标系统的机器学习架构，以构建与原始机器学习架构具有相同功能的影子或替代模型（Barreno 等人，2006 年；Tramer 等人，2016 年； Wu 等人 2016 年；Papernot 等人 2017 年；Shi 等人 2017 年；Shi 等人 2018b)。这对应于白盒或黑盒攻击，具体取决于对手是否可以使用深度神经网络结构等机器学习模型。对于黑盒攻击，对手使用大量样本查询目标分类器并记录标签。然后，它使用这个标记数据作为自己的训练数据来训练一个功能等效（即统计相似）的深度学习分类器，即代理模型。
b. 成员推理攻击：对手旨在确定给定数据样本是否是训练数据的成员，即给定数据样本是否已用于训练感兴趣的机器学习算法（Nasr 等人，2018 年；Song 等人，2018 年）。 2018 年；Jia 等人 2019 年；莱诺和弗雷德里克森 2020 年）。Membership inference attack 是基于过度拟合的分析，以检查机器学习算法是否针对特定数据类型（例如特定类型的图像）进行了训练。通过了解训练机器学习算法分类的数据类型，对手可以更成功地设计后续攻击。

## 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|ECON90022

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|A Single Leader Multi-followers Stackelberg Game

In a single leader multiple followers (SLMF) game (Basar and Olsder 1999), the leader makes its optimal decision prior to the decisions of multiple followers. The Stackelberg game played by the leader is:
$$\begin{array}{ll} \min _{s^{\ell}, s^} & F\left(s^{\ell}, s^\right) \ \text { s.t. } & G\left(s^{\ell}, s^\right) \leq 0 \ & H\left(s^{\ell}, s^\right)=0 \end{array}$$
where $F$ is the leader’s objective function, constrained by $G$ and $H ; s^{\prime}$ is the leader’s decision and $s^$ is in the set of the optimal solutions of the lower level problem: $$s^ \in\left{\begin{array}{ll} \underset{s_i}{\operatorname{argmin}} & f_i\left(s^t, s_i\right) \ \text { s.t. } & g_i\left(s^{\ell}, s_i\right) \leq 0 \ & h_i\left(s^{\ell}, s_i\right)=0 \end{array}\right} \quad \forall i=1, \ldots, m$$
where $m$ is the number of followers, $f_i$ is the $i^{\text {th }}$ follower’s objective function constrained by $g_l$ and $h_i$

For the sake of simplicity, we assume the followers are not competing among themselves. This is usually a valid assumption in practice since adversaries rarely affect each other through their actions. In a Bayesian Stackelberg game, the followers may have many different types and the leader does not know exactly the types of adversaries it may face when solving its optimization problem. However, the distribution of the types of adversaries is known or can be inferred from past experience. The followers’ strategies and payoffs are determined by the followers’ types. The followers play their optimal responses to maximize the payoffs given the leader’s strategy. The Stackelberg equilibrium includes an optimal mixed strategy of the learner and corresponding optimal strategies of the followers.

Problem Definition Given the payoff matrices $R^{\ell}$ and $R^f$ of the leader and the $m$ followers of $n$ different types, find the leader’s optimal mixed strategy given that all followers know the leader’s strategy when optimizing their rewards. The leader’s pure strategies consist of a set of generalized linear learning models $\langle\phi(x), w\rangle$ and the followers’ pure strategies include a set of vectors performing data transformation $x \rightarrow x+\delta x$.

Given the payoff matrices $R^{\ell}$ and $R^f$ discussed in Section 13.5.3.3 (as shown in Table 13.6), let $r^f$ denote the follower’s maximum payoff, $\mathcal{L}$ and $\mathcal{F}$ denote the indices of the pure strategies in the leader’s policy $s^{\ell}$ and the follower’s policy $s^f$.

## 经济代写|博弈论代写Game Theory代考|Real Datasets

For the two real datasets we again use spam base (UCI Machine Learning Repository 2019) and web spam (LIBSVM Data 2019). The learning tasks are binary classification problems, differentiating spam or webspam from legitimate e-mail or websites.

Spambase Dataset Recall that in the spam base dataset, the task is to differentiate spam from legitimate e-mail. There are 4601 e-mail messages in the dataset including approximately 1800 spam messages. The dataset has 57 attributes and one class label. Training and test datasets are separate. The results are averaged over 10 random runs. The detailed results are shown in Table 13.8. The $f_{\mathrm{a}}=0$ column is left out for the same reason as explained earlier on the artificial datasets.

This dataset serves as an excellent example to demonstrate the power of the mixed strategy. In the cases where $p=0.1$, that is, when we assume legitimate e-mail is modified $10 \%$ of the time (while spam is always modified), the two equilibrium predictors Equi $^{* 1}$ and Equi ${ }^{* 2}$ exhibit very stable performance in terms of predictive accuracy. Their error rates fluctuate slightly at 0.37 regardless of how aggressively the test data has been modified. On the other hand, $S \mathrm{SM}^{* 1}$ and $S_{V M}{ }^{* 2}$ significantly outperform the equilibrium predictors Equi ${ }^{* 1}$ and Equi $^{* 2}$ and the invariant SVM when $f_{\mathrm{a}} \leq 0.5$. However, the performance of SVM ${ }^{* 1}$ and SVM ${ }^{* 2}$ dropped quickly as the attack gets more intense $\left(f_{\mathrm{a}}>0.5\right)$, much poorer than the equilibrium predictors. The mixed strategy, although not the best, demonstrates superb performance by agreeing with the winning models the majority of the time. The standard SVM has similar performance to the equilibrium predictors, behaving poorly as the attack gets intense. When $p=0.5$, that is, when legitimate e-mail is modified half of the time while all spam is modified, equilibrium predictors still demonstrate very stable performance while the performance of the equilibrium predictors Equi $^{* 1}$ and Equi ${ }^{* 2}$ deteriorates sharply right after the attack factor increases to 0.3 . The mixed strategy, again not the best predictor, demonstrates the most consistent performance among all the predictors given any attack intensity levels.

We also tested the case where the attack factor $f_{\mathrm{a}} \in(0,1)$ is completely random under uniform distribution for each attacked sample on this dataset. The probability of negative data being attacked increases gradually from 0.1 to 0.9 . The results are illustrated in Figure 13.11. Again, we observe similar behavior of all the predictors: stable equilibrium predictors, $\mathrm{SVM}^{* 1}, \mathrm{SVM}^{* 2}$, and SVM progressively deteriorating as $p$ increases. The mixed strategy, although weakened as more negative data is allowed to be modified, consistently lies in between the equilibrium predictors and the $S V M$ predictors.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|Real Datasets

Spambase 数据集 回想一下，在垃圾邮件基础数据集中，任务是区分垃圾邮件和合法电子邮件。数据集中 有 4601 封电子邮件，包括大约 1800 封垃圾邮件。该数据集有 57 个属性和一个类标签。训练和测试数据 集是分开的。结果取 10 次随机运行的平均值。详细结果如表13.8所示。这 $f_{\mathrm{a}}=0$ 出于与前面在人工数据 集上解释的相同原因，列被遗漏了。

## 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|ECON2070

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|Equilibrium Strategy Determination

In this game, the defender is considered the controller as it can select the actions which move the game from a state to another. The defender will also control when to apply the MTD, i.e. it will determine the duration between the time steps of the stochastic game. Assuming that the attacker has enough power, it can complete the brute-force attack in time $t_i$ for $i=1,2, \ldots, N$ for each one of the encryption techniques. Then, the defender should choose the time step $t$ to take the next action as follows:
$$t<\min \left(t_i\right), \quad i=1,2, \ldots, N .$$
By doing this, the defender can make sure that it takes a timely action before the attacker succeeds in revealing one of the keys.
The accumulated utility of player $i$ at state $s$ will be
$$\Phi_i(\boldsymbol{f}, \boldsymbol{g}, s)=\sum_{t=1}^{\infty} \beta^{t-1} \cdot U_i\left(f\left(s_t\right), g\left(s_t\right), s_t\right),$$
where $\boldsymbol{f}$ and $\boldsymbol{g}$ are the strategies adopted by the defender and attacker, respectively. The strategy specifies a vector of actions to be chosen at each of the states, e.g. $\boldsymbol{f}=\left[f\left(s_1\right), \ldots, f\left(s_K\right)\right]$ for all the $K$ states. Actions $f\left(s_t\right)$ and $g\left(s_t\right)$ are the actions chosen at $s_t$, which is the state of the game at time $t$, according to strategies $\boldsymbol{f}, \boldsymbol{g}$. State $s_t \in \mathcal{\delta}$ is determined by the defender’s action at time $t-1$. The game is assumed to start at a specific state $s=s_1$. Note that the utility in (10.4) is always bounded at infinity due to the fact that $0<\beta<1$.

When designing the bimatrix, the defender needs to calculate the accumulated utility when choosing each pure strategy against all of the attacker’s pure strategies. The defender, as a controller, can know the next state resulting from its actions, and, thus, it sums the utilities in all states using the discount factor $\boldsymbol{\beta}$. Let $\boldsymbol{X}$ be the defender’s accumulated utility matrix for all defender’s pure strategies’ permutations and all attacker’s pure strategies’ permutations. We let $\boldsymbol{F}{\boldsymbol{\bullet} \bullet}=\left[\boldsymbol{f}_1, \boldsymbol{f}_2, \ldots, \boldsymbol{f}{K^k}\right]$ be a matrix of all defender’s pure strategies’ permutation where each row represents actions in this strategy and similarly $\boldsymbol{G}{i \bullet}=\left[\boldsymbol{g}_1, \boldsymbol{g}_2, \ldots, \boldsymbol{g}{N^k}\right]$ the matrix of all attacker’s pure strategies’ permutation.

## 经济代写|博弈论代写Game Theory代考|Simulation Results and Analysis

For our simulations, we choose a system that uses two encryption techniques with two different keys per technique. Thus, the number of system states is four and the defender has four actions in each state. For the bimatrix, the attacker has $2^4=16$ different strategy permutations and the defender has $4^4=256$ different strategy permutations. The power values are set to 1 and 3 to pertain to the ratio between the power consumption in the two different encryption techniques. These values are the same for both players. We set $R_1$ and $R_2$ to be 10 and 5 depending on the opponent’s actions. We choose these values to be higher than the power values in order for the utilities to be positive. The transition reward is set to 5 and 10 for switching to another state defined by another key or another technique, respectively.

First, we run simulations when there is no transition cost, $q=0$. The equilibrium strategies for both the attacker and defender are shown in Table 10.2. Note that actions $a_1, a_2$ represent the selection of two keys for the same encryption technique and actions $a_3, a_4$ represent two keys for another technique. Table 10.1 shows the probabilities over all actions for each player. These probabilities show how players should select actions in every state. For the defender, if it starts in state $s_3$, then it should move to state $s_1$ with the highest probability and move to state $s_2$ with a very similar probability. This is because the defender will change the technique and so gets a higher transition reward. We can see that the probability of moving to the same state is always very low and can reach 0 as in state $s_1$. The probability of moving to a state that has a similar encryption key is always less than that of moving to a state with different technique as the transition reward will be lower. For the attacker, the probability of attacking the same technique that is used in the current state is always higher than attacking any other technique.

In Figure 10.3, we show the effect of the discount factor on the defender’s utility at equilibrium in every state. First, we can see that all utility values at all states increase as the discount factor increases. This is due to the fact that increasing the discount factor will make the defender care more about future rewards thus choosing the actions that will increase these future rewards. Figure 10.3 also shows that the defender’s values at states 1 and 2 are higher than at states 3 and 4 . This because states 1 and 2 adopt the first encryption technique which uses less power than the encryption technique used in states 3 and 4 . The difference mainly arises in the first state before switching to other states and applying the discount factor. Clearly, changing the discount factor has a big effect on changing the equilibrium strategy, and, thus, the game will move between states with different probabilities resulting in a different accumulated reward.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|Equilibrium Strategy Determination

$$t<\min \left(t_i\right), \quad i=1,2, \ldots, N$$

$$\Phi_i(\boldsymbol{f}, \boldsymbol{g}, s)=\sum_{t=1}^{\infty} \beta^{t-1} \cdot U_i\left(f\left(s_t\right), g\left(s_t\right), s_t\right)$$

## 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|ECON3503

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|Numerical Example

Consider a game that has two states. In an MTD scenario, the defender will have two actions representing selecting between each of these two states. Similarly, the attacker is assumed to have two actions of attacking profiles (both can be applied at either state).

A defender’s strategy can then be given as $\boldsymbol{f}=[a b]$, where $a$ is the action of moving to state $s_1$ and $b$ is the action of moving to state $s_2$.
The defender’s and attacker’s strategies’ permutation will be given as:
$$\boldsymbol{F}=\left[\begin{array}{ll} 1 & 1 \ 1 & 2 \ 2 & 1 \ 2 & 2 \end{array}\right] \boldsymbol{G}=\left[\begin{array}{ll} 1 & 1 \ 1 & 2 \ 2 & 1 \ 2 & 2 \end{array}\right]$$
where $\boldsymbol{F}$ and $\boldsymbol{G}$ represent the defender’s and the attacker’s permutations, respectively.
Since each player has four different permutations, each of the formulated matrices (bimatrix) will be of size $4 \cdot 4$ representing all the possible combinations of the players’ permutations. The elements of these matrices will be the accumulated utilities for each player resulting from starting at combination and considering all the future transitions.

Now suppose that the mixed Nash equilibrium for the bimatrix game, calculated from any numerical algorithm such as Lemke and Howson (1964), is given by:
$$\boldsymbol{x}^=\left[\begin{array}{l} 0.2 \ 0.1 \ 0.3 \ 0.4 \end{array}\right] \boldsymbol{y}^=\left[\begin{array}{l} 0.0 \ 0.1 \ 0.4 \ 0.5 \end{array}\right]$$

where each row in $\boldsymbol{x}^$ and $\boldsymbol{y}^$ represents the players’ probabilities of selecting a strategy in $\boldsymbol{F}$ and $\boldsymbol{G}$, respectively.

Finally, the stochastic game equilibrium strategies can be calculated by summing the probabilities of choosing each action over the corresponding states. For example, the defender can choose action 1 at state 1 twice in $\boldsymbol{F}$ with probabilities 0.2 and 0.1 (from $\boldsymbol{x}^$ ). When summed, the defender knows that when the game is at state 1 , it will choose action 1 with a probability 0.3 . Following the same approach, we can compute the full equilibrium matrices as follows: $$\boldsymbol{E}^=\left[\begin{array}{ll} 0.3 & 0.5 \ 0.7 & 0.5 \end{array}\right] \boldsymbol{H}^=\left[\begin{array}{ll} 0.1 & 0.4 \ 0.9 & 0.6 \end{array}\right]$$ where $\boldsymbol{E}^$ and $\boldsymbol{H}^*$ are the defender’s and the attacker’s equilibrium solutions, respectively, and that the rows of the matrices represent players’ actions and the columns represent the game states.

## 经济代写|博弈论代写Game Theory代考|A Case Study for Applying Single-Controller

Consider a wireless sensor network that consists of a BS and a number of wireless nodes. The network is deployed for sensing and collecting data about some phenomena in a given geographic area. Sensors will collect data and use multi-hop transmissions to forward this data to a central receiver or BS. The multiple access follows a slotted Aloha protocol. Time is divided into slots and the time slot size equals the time required to process and send one packet. Sensor nodes are synchronized with respect to time slots. We assume that nodes are continuously working and so every time slot there will be data that must be sent to the BS.

All packets sent over the network are assumed to be decrypted using a given encryption technique and a previously shared secret key. All the nodes in the system are pre-programmed with a number of encryption techniques along with a number of encryption keys per technique, as what is typically done in sensor networks (Casola et al. 2013). The BS chooses a specific encryption technique and key by sending a specific control signal over the network including the combination it wants to use. We note that the encryption technique and key sizes should be carefully selected in order not to consume a significant amount of energy when encrypting or decrypting packets. Increasing the key size will increase the amount of consumed energy particularly during the decryption (Lee et al. 2010). Since the BS is mostly receiving data, it spends more time decrypting packets rather than encrypting them and, thus, it will be highly affected by key size selection.

In our model, an eavesdropper is located in the communication field of the BS and it can listen to packets sent or received by the BS. As packets are encrypted, the attacker will seek to decrypt the packets it receives in order to get information. The attacker knows the encryption techniques used in the network and so it can try every possible key on the received packets until getting useful information. This technique is known as brute-force attack.

The idea of using multiple encryption techniques was introduced in Casola et al. (2013). However, in this work, each node individually selects one of these technique to encrypt transmitted packets. The receiving node can know the used technique by a specific field in the packet header. Large encryption keys were used which require a significant amount of power to be decrypted. Nonetheless, these large keys are highly unlikely to be revealed using a brute-force attack in a reasonable time. Here, we propose to use small encryption keys to save energy and, in conjunction with that, we enable the BS to change the encryption method in a way that reduces the chance that the encryption key is revealed by the attacker. This is the main idea behind MTD. In this model, the encryption key represents the attack surface, and by changing the encryption method, the BS will make it harder for the eavesdropper to reveal the key and get the information from the system. Naturally, the goals of the eavesdropper and the BS are not aligned. On the one hand, the BS wants to protect the data sent over the network by changing encryption method. On the other hand, the attacker wants to reveal the used key in order to get information. To understand the interactions between the defender and the attacker, one can use game theory to study their behavior in this MTD scenario. The problem is modeled as a game in which the attacker and the defender are the players. As the encryption method should be changed over time and depending on the attacker’s actions, we must use a dynamic game.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|Numerical Example

$$\boldsymbol{x}^{=}\left[\begin{array}{llll} 0.2 & 0.1 & 0.3 & 0.4 \end{array}\right] \boldsymbol{y}=\left[\begin{array}{llll} 0.0 & 0.1 & 0.4 & 0.5 \end{array}\right]$$

## 经济代写|博弈论代写Game Theory代考|A Case Study for Applying Single-Controller

Casola 等人介绍了使用多种加密技术的想法。(2013)。然而，在这项工作中，每个节点单独选择其中一种技术来加密传输的数据包。接收节点可以通过包头中的特定字段知道所使用的技术。使用了需要大量能量才能解密的大型加密密钥。尽管如此，这些大密钥极不可能在合理的时间内使用暴力攻击来泄露。在这里，我们建议使用小的加密密钥来节省能源，与此同时，我们使 BS 能够以一种降低加密密钥被攻击者泄露的机会的方式改变加密方法。这是 MTD 背后的主要思想。在这个模型中，加密密钥代表攻击面，通过改变加密方式，BS 将使窃听者更难泄露密钥并从系统中获取信息。自然地，窃听者和 BS 的目标并不一致。一方面，BS 想通过改变加密方法来保护通过网络发送的数据。另一方面，攻击者想要泄露使用的密钥以获取信息。要了解防御者和攻击者之间的交互，可以使用博弈论来研究他们在此 MTD 场景中的行为。该问题被建模为一个游戏，其中攻击者和防御者是玩家。由于加密方法应该随着时间的推移而改变，并且取决于攻击者的行为，我们必须使用动态游戏。窃听者和 BS 的目标不一致。一方面，BS 想通过改变加密方法来保护通过网络发送的数据。另一方面，攻击者想要泄露使用的密钥以获取信息。要了解防御者和攻击者之间的交互，可以使用博弈论来研究他们在此 MTD 场景中的行为。该问题被建模为一个游戏，其中攻击者和防御者是玩家。由于加密方法应该随着时间的推移而改变，并且取决于攻击者的行为，我们必须使用动态游戏。窃听者和 BS 的目标不一致。一方面，BS 想通过改变加密方法来保护通过网络发送的数据。另一方面，攻击者想要泄露使用的密钥以获取信息。要了解防御者和攻击者之间的交互，可以使用博弈论来研究他们在此 MTD 场景中的行为。该问题被建模为一个游戏，其中攻击者和防御者是玩家。由于加密方法应该随着时间的推移而改变，并且取决于攻击者的行为，我们必须使用动态游戏。要了解防御者和攻击者之间的交互，可以使用博弈论来研究他们在此 MTD 场景中的行为。该问题被建模为一个游戏，其中攻击者和防御者是玩家。由于加密方法应该随着时间的推移而改变，并且取决于攻击者的行为，我们必须使用动态游戏。要了解防御者和攻击者之间的交互，可以使用博弈论来研究他们在此 MTD 场景中的行为。该问题被建模为一个游戏，其中攻击者和防御者是玩家。由于加密方法应该随着时间的推移而改变，并且取决于攻击者的行为，我们必须使用动态游戏。

## 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|ECON3503

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|Hypergames on Graphs

A hypergame models the situation where different players perceive their interaction with other players differently and consequently play different games in their own minds depending on their perception. We consider the case where the difference in perception arises because of incomplete and potentially incorrect information. For instance, suppose a subset of nodes in the network are honeypots, the attacker may mistake these to be true hosts. We formulate a hypergame to model the interaction between the defender and the attacker given asymmetric information.
First, let’s review the definition of hypergames.
Definition 6.2 (Hypergame; Bennett 1980; Vane 2000) Given two players, a game perceived by player 1 is denoted by $\mathcal{G}_1$, and a game perceived by player 2 is denoted by $\mathcal{G}_2$. A level-1 hypergame is defined as a tuple:
$$\mathcal{H} \mathcal{G}^1=\left\langle\mathcal{G}_1, \mathcal{G}_2\right\rangle$$
In a level-1 hypergame, none of the player’s is aware of other player’s perception.
When one player becomes aware of the other player’s (mis)perception, the interaction is captured by a level- 2 two-player hypergame, defined as a tuple:
$$\mathcal{H} \mathcal{G}^2=\left\langle\mathcal{H} \mathcal{G}^1, \mathcal{G}_2\right\rangle$$
where P1 perceives the interaction as a level-1 hypergame (as P1 is aware of P2’s game $\mathcal{G}_2$ in addition to his own) and $\mathrm{P} 2$ perceives the interaction as the game $\mathcal{G}_2$.

We refer to the games $\mathcal{G}_1$ (resp., $\mathcal{G}_2$ ) as P1’s (resp., P2’s) perceptual game in level-1 hypergame, and $\mathcal{H} \mathcal{G}^1$ as P1’s perceptual game in level-2 hypergame. As P2 is not aware that she might be misperceiving the game, her perceptual game in level-2 hypergame is still $\mathcal{G}_2$.

In general, if P1 computes his strategy by solving an $(m-1)$ th level hypergame and P2 computes her strategy using an $n$th level hypergame with $n<m$, then the resulting hypergame is said to be a level- $m$ hypergame given as:
$$\mathcal{H} \mathcal{G}^m=\left\langle\mathcal{H} \mathcal{G}_1^{m-1}, \mathcal{H} \mathcal{G}_2^n\right\rangle$$

## 经济代写|博弈论代写Game Theory代考|Synthesis of Reactive Defense Strategies with Cyber Deception

When the attacker has a one-sided misperception of labeling function, as defined in Definition 6.3, the defender might strategically utilize this misperception to deceive the attacker into choosing a strategy that is advantageous to the defender. To understand when the defender might have such a deceptive strategy and how to compute it, we study the solution concept of hypergame.

Solution Approach A hypergame $\mathcal{H} \mathcal{G}^2=\left\langle\mathcal{H} \mathcal{G}^1, \mathcal{G}_2\right\rangle$ is defined using two games, namely $\mathcal{G}_1$ and $\mathcal{G}_2$. Under one-sided misperception of labeling function, defender is aware of both games. Therefore, to synthesize a deceptive strategy, the defender must take into account the strategy that the attacker will use, based on her misperception. That is, the defender must solve two games: game $\mathcal{G}_2$ to identify the set of states in the game transition system $G_2 \otimes \mathcal{A}$ that the attacker perceives as winning for her under labeling function $L_2$ and game $\mathcal{G}_1$ to identify the set of states in the game transition system $G_1 \otimes \mathcal{A}$ that are winning for the defender under (ground-truth) labeling function $L=L_1$. After solving the two games, the defender can integrate the solutions to obtain a set of states, at which the attacker makes mistakes due to the difference between $L_2$ and $L$. Let us introduce a notation to denote these sets of winning states.

• $\mathcal{G}_1$ : P1’s winning region is Win $\operatorname{Win}_1 \subseteq S \times Q$ and P2’s winning region is Win $_2 \subseteq S \times Q$.
• $\mathcal{G}_2: \mathrm{P} 1$ ‘s winning region is $\mathrm{Win}_1^{\mathrm{P2}} \subseteq S \times Q$ and P2’s winning region is $\mathrm{Win}_2^{\mathrm{P2}} \subseteq S \times Q$.
Figure 6.7 provides a conceptual representation partitions of the state space of a game transition system. Due to misperception, the set of states are partitioned into the following regions:
• Win $\mathrm{W}_1$ : is a set of states from which P1 can ensure satisfaction of security objectives, even if P2 has complete and correct information. Thus, P1 can take the winning strategy $\pi_1$.
• $\mathrm{Win}_1^{\mathrm{P} 2} \cap \mathrm{Win}_2$ is a set of states where P2 is truly winning but perceives the states to be losing for her due to misperception. Thus, P2 may either give up the attack mission or play randomly.
• $\mathrm{Win}_2^{\mathrm{P2}} \cap \mathrm{Win}_2$ is a set of states in which P2 is truly winning and perceives those states to be winning. In this scenario, she will carry out the winning strategy $\pi_2^{\mathrm{P2}}$. However, this strategy can be different from the true winning strategy $\pi_2$ that P2 should have played if she had complete and correct information. This difference creates unique opportunities for P1 to enforce security of the system.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|Hypergames on Graphs

$$\mathcal{H \mathcal { G } ^ { 1 }}=\left\langle\mathcal{G}_1, \mathcal{G}_2\right\rangle$$

$$\mathcal{H} \mathcal{G}^2=\left\langle\mathcal{H} \mathcal{G}^1, \mathcal{G}_2\right\rangle$$

$$\mathcal{H} \mathcal{G}^m=\left\langle\mathcal{H} \mathcal{G}_1^{m-1}, \mathcal{H} \mathcal{G}_2^n\right\rangle$$

## 经济代写|博弈论代写Game Theory代考|Synthesis of Reactive Defense Strategies with Cyber Deception

• $\mathcal{G}_1: \mathrm{P} 1$ 的获胜区域是WinWin $\operatorname{Win}_1 \subseteq S \times Q \mathrm{P} 2$ 的获胜区域是 $\operatorname{Win}_2 \subseteq S \times Q$.
• $\mathcal{G}_2: \mathrm{P} 1$ 的获胜区域是 $\mathrm{Win}_1^{\mathrm{P} 2} \subseteq S \times Q \mathrm{P} 2$ 的获胜区域是 $\mathrm{Win}_2^{\mathrm{P} 2} \subseteq S \times Q$. 图 6.7 提供了游戏转换系统状态空间的概念表示分区。由于误解，状态集被划分为以下区域:
• 赢 $W_1$ : 是一组状态，即使 $P 2$ 拥有完整和正确的信息，P1 也可以确保满足安全目标。因此，P1 可以釆取必胜策略 $\pi_1$.
• $\operatorname{Win}_1^{\mathrm{P2}} \cap \mathrm{Win}$ 是一组状态，其中 $P 2$ 真正获胜，但由于误解而认为这些状态对她来说是失败 的。因此，P2 要么放弃攻击任务，要么随机游玩。
• $\mathrm{Win}_2^{\mathrm{P} 2} \cap \mathrm{Win}_2$ 是 $\mathrm{P} 2$ 真正获胜并认为这些状态获胜的一组状态。在这种情况下，她将执行制胜 战略 $\pi_2^{\mathrm{P} 2}$. 但是，此策略可能与真正的获胜策略不同 $\pi_2$ 如果她有完整和正确的信息， $P 2$ 应该玩。这 种差异为 P1 创造了独特的机会来加强系统的安全性。

## 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|ECON90022

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|A Theory of Hypergames on Graphs

A game arena is a transition system with labels assigned to the states. It captures different configurations of the network and the actions that the attacker and the defender may use to change the current configuration. A configuration of system is a set of state variables that jointly define the current state of the system. For instance, a state variable may be a collection of the current host compromised by the attacker, IP addresses of different hosts over the network, an enumeration of services running over each host, or a list of users currently accessing the hosts with their privileges (root, user, and none). Suppose that there are $n$ state variables and we denote the $i$ th state variable as $X_i$, then the domain of a state space can be given by $S=X_1 \times X_2 \times \cdots \times X_n$. Given this notion of state, we formally define a game arena as follows:

Definition 6.1 (Arena) A turn-based, deterministic game arena between two players P1 (defender, pronoun “he”) and P2 (attacker, pronoun “she”) is a tuple
$$G=\langle S, A, T, \mathcal{A} \mathcal{P}, L\rangle,$$
whose components are defined as follows:

• $S=S_1 \cup S_2$ is a finite set of states partitioned into two sets $S_1$ and $S_2$. At a state in $S_1$, P1 chooses an action and at a state in $S_2, \mathrm{P} 2$ selects an action.
• $A=A_1 \cup A_2$ is the set of actions. $A_1$ (resp., $A_2$ ) is the set of actions for P1 (resp., P2);
• $T:\left(S_1 \times A_1\right) \cup\left(S_2 \times A_2\right) \rightarrow S$ is a deterministic transition function that maps a state-action pair to the next state.
• $\mathcal{A P}$ is the set of atomic propositions.
• $L: S \rightarrow 2^{\mathcal{A P}}$ is the labeling function that maps each state $s \in S$ to a set $L(s) \subseteq \mathcal{A} \mathcal{P}$ of atomic propositions that evaluate to true at that state.

## 经济代写|博弈论代写Game Theory代考|Specifying the Security Properties in LTL

We consider qualitative formal specifications for defender and attacker objectives. Different from quantitative utility functions in terms of costs, qualitative logic formulas capture hard security constraints that the network defense system must satisfy.

The defender has two types of goals, namely (i) operational objectives, such as the services should eventually be available to the legitimate users, and (ii) defense objectives, such as the attacker should never be able to compromise servers with sensitive information. However, the intention of attacker is often unknown. Thus, we consider the worst-case scenario where the attacker’s objective is to violate the security goal of the defender.
We choose to express the security goal of the defender using LTL (Manna and Pnueli 1992). LTL allows us to express the security properties of system with respect to time. We shall now present the formal syntax and semantics of LTL and then discuss several examples.

Let $\mathcal{A P}$ be a set of atomic propositions. Linear Temporal Logic (LTL) has the following syntax,
$$\varphi:=\mathrm{T}|\perp| p|\varphi| \neg \varphi\left|\varphi_1 \wedge \varphi_2\right| \bigcirc \varphi \mid \varphi_1 \mathrm{U} \varphi_2$$
where

• $T, \perp$ represent universally true and false, respectively.
• $p \in \mathcal{A} \mathcal{P}$ is an atomic proposition.
• $O$ is a temporal operator called the “next” operator (see semantics below).
• $\mathrm{U}$ is a temporal operator called the “until” operator (see semantics below).
Let $\Sigma:=2^{\mathcal{A} P}$ be the finite alphabet. Given a word $w \in \Sigma^\omega$, let $w[i]$ be the $i$ th element in the word and $w[i \ldots]$ be the subsequence of $w$ starting from the ith element. For example, $w=a b c, w[0]=a$, and $w[1 \ldots]=b c$. Formally, we have the following definition of the semantics:
• $w \vDash p$ if $p \in w[0]$;
• $w \vDash \neg p$ if $p \notin w[0]$
• $w \vDash \varphi_1 \wedge \varphi_2$ if $w \vDash \varphi_1$ and $w \vDash \varphi_2$.
• $w \vDash \bigcirc \varphi$ if $w[1 \ldots] \vDash \varphi$.
• $w \vDash \varphi \cup \psi$ if $\exists i \geq 0, w[i \ldots] \vDash \psi$ and $\forall 0 \leq j<i, w[j \ldots] \vDash \varphi$.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|A Theory of Hypergames on Graphs

$$G=\langle S, A, T, \mathcal{A} \mathcal{P}, L\rangle$$

• $S=S_1 \cup S_2$ 是分成两组的有限状态集 $S_1$ 和 $S_2$. 在一个状态 $S_1$ ，P1 选择一个动作并且处于一个状 态 $S_2, \mathrm{P} 2$ 选择一个动作。
• $A=A_1 \cup A_2$ 是动作集。 $A_1$ (分别， $A_2$ ) 是 P1 (resp., P2) 的动作集；
• $T:\left(S_1 \times A_1\right) \cup\left(S_2 \times A_2\right) \rightarrow S$ 是将状态-动作对映射到下一个状态的确定性转换函数。
• $\mathcal{A} \mathcal{P}$ 是原子命题的集合。
• $L: S \rightarrow 2^{\mathcal{A P}}$ 是映射每个状态的标签函数 $s \in S$ 一组 $L(s) \subseteq \mathcal{A} \mathcal{P}$ 在该状态下评估为真的原子命 题。

## 经济代写|博弈论代写Game Theory代考|Specifying the Security Properties in LTL

$$\varphi:=\mathrm{T}|\perp| p|\varphi| \neg \varphi\left|\varphi_1 \wedge \varphi_2\right| \bigcirc \varphi \mid \varphi_1 \mathrm{U} \varphi_2$$

• $T, \perp$ 分别代表普遍正确和普遍错误。
• $p \in \mathcal{A} \mathcal{P}$ 是一个原子命题。
• $O$ 是一个称为“下一个”运算符的时间运算符（参见下面的语义）。
• U是一个称为”until”运算符的时间运算符 (参见下面的语义)。
让 $\Sigma:=2^{\mathcal{A P}}$ 是有限的字母表。给了一个字 $w \in \Sigma^\omega$ ，让 $w[i]$ 成为 $i$ 单词中的第 th 个元素和 $w[i \ldots]$ 是的后续 $w$ 从第 $\mathrm{i}$ 个元素开始。例如， $w=a b c, w[0]=a$ ，和 $w[1 \ldots]=b c$. 形式上， 我们有以下语义定义:
• $w \models p$ 如果 $p \in w[0]$;
• $w \models \neg p$ 如果 $p \notin w[0]$
• $w \models \varphi_1 \wedge \varphi_2$ 如果 $w \models \varphi_1$ 和 $w \models \varphi_2$.
• $w \models \bigcirc \varphi$ 如果 $w[1 \ldots] \models \varphi$.
• $w \models \varphi \cup \psi$ 如果 $\exists i \geq 0, w[i \ldots] \models \psi$ 和 $\forall 0 \leq j<i, w[j \ldots] \models \varphi$.

## 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|ECON40010

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|Solution Concepts

Since EFGs can be converted into NFGs, the solution concepts introduced in Section 2.3 .1 still apply. Also, there are other solution concepts specifically for EFGs. We briefly introduce some of them at a high level.

One solution concept is subgame perfect equilibrium (or subgame perfect Nash equilibrium, SPE in short). It is a refinement of NE. This means the set of SPE is a subset of NE. A strategy profile is an SPE equilibrium if, in every subgame of the original game, the strategies remain an NE. A subgame is defined as the partial tree consisting of a node and all its successors, with the requirement that the root node of the partial tree is the only node in its information set. This means if the players play a small game that is part of the original game, they still have no incentive to deviate from their current strategy. A subgame can be a subgame of another subgame rooted at its ancestor node. A finite EFG with perfect recall always has an SPE. To find an SPE in a finite game with perfect information, one can use backward induction as all the information sets are singletons. First, consider the smallest subgames rooted at a parent node of a terminal node. After solving these subgames by determining the action to maximize the acting player’s expected utility, one can solve a slightly larger subgame whose subgames have already been solved, assuming the player will play according to the smaller subgames’ solution. This process continues until the original game is solved. The resulting strategy is an SPE. For games of imperfect or incomplete information, backward induction cannot be applied as it will require reasoning about the information sets with more than one node. The SPE concept can also be defined through the one-shot deviation principle: a strategy profile is an SPE if and only if no player can gain any utility by deviating from their strategy for just one decision point and then reverting back to their strategy in any subgame.

In the example game in Figure 2.1, if we remove the dashed box, the game becomes one with perfect information, and it has two subgames in addition to the original game, with the root nodes being nodes 2 and 3 respectively. In this new game, Player 1 choosing $L$ and Player 2 choosing $r$ at both nodes 2 and 3 is an NE but not an SPE, because, in the subgame rooted at node 3, Player 2 choosing $\mathrm{r}$ is not the best action. In contrast, Player 1 choosing $\mathrm{R}$ and Player 2 choosing $\mathrm{r}$ at node 2 , 1 at node 3 is an SPE.

Another solution concept is sequential equilibrium (Kreps and Wilson 1982), which is a further refinement of SPE. A sequential equilibrium consists of not only the players’ strategies but also the players’ beliefs for each information set. The belief describes a probability distribution on the nodes in the information set since the acting player cannot distinguish them when they play the game. Unlike SPE which only considers the subgames rooted from a node that is the only element in its information set, the sequential equilibrium requires that the players are sequentially rational and takes the best action in terms of expected utility concerning the belief at every information set even if it is not a singleton.

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

Consider a two-player NFG with finite actions. The two players choose an action in their action set simultaneously without knowing the other player’s strategy ahead of time. In an NE of the game, each player’s strategy is a best response to the other player’s strategy. But what if one player’s strategy is always known to other players ahead of time? This kind of role asymmetry is depicted in a Stackelberg game.

In a Stackelberg game, one player is the leader and chooses her strategy first, and the other players are followers who observe the leader’s strategy and then choose their strategies. The leader has to commit to the strategy she chooses, i.e. she cannot announce a strategy to the followers and play a different strategy when the game is actually played. She can commit to either a pure strategy or a mixed strategy. If she commits to a mixed strategy, the followers can observe her mixed strategy but not the realization of sampled action from this mixed strategy. A Stackelberg game can be described in the same way as before, with a tuple $(\mathcal{N}, \mathcal{A}, u)$ for games in normal form and $(\mathcal{N}, \mathcal{A}, \mathcal{H}, \mathcal{Z}, \chi, \rho, \sigma, u)$ for games in extensive form, with the only difference that one of the players (usually Player 1 ) is assumed to be the leader.

At first sight, one may think that the followers have an advantage as they have more information than the leader when they choose their strategies. However, it is not the case. Consider the Football vs. Concert game again. If the row player is the leader and commits to choosing the action $\mathrm{F}$, a rational column player will choose action $\mathrm{F}$ to ensure a positive payoff. Thus, the row player guarantees herself a utility of 2, the highest utility she can get in this game, through committing to playing pure strategy $\mathrm{F}$. In the real world, it is common to see similar scenarios. When a couple is deciding what to do during the weekend, if one of them has a stronger opinion and insists on going to the activity he or she prefers, they will likely choose that activity in the end. In this example game, the leader ensures a utility that equals the best possible utility she can get in an NE. In some other games, the leader can get a utility higher than any NE in the game by committing to a good strategy. In the example game in Table 2.5 (Up, Right) is the only NE, with the row player’s utility being 4 . However, if the row player is the leader and commits to a uniform random strategy, the column player will choose Left as a best response, yielding an expected utility of 5 for the row player. This higher expected utility for the row players shows the power of commitment.

# 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|Normal-Form Games

In this section, we introduce normal-form games (NFGs) and formally introduce the solution concepts we mentioned in Section 2.2. Most of the notations and definitions are adapted from Leyton-Brown and Shoham 2008. The normal form is the most basic form of games to represent players’ interactions and strategy space. An NFG captures all possible combinations of actions or strategies for the players and their corresponding payoffs in a matrix, or multiple matrices for more than two players. A player can choose either a pure strategy that deterministically selects a single strategy or play a mixed strategy that specifies a probability distribution over the pure strategies. The goal for all players is to maximize their expected utility. Formally, a finite, $N$-person NFG is described by a tuple $(\mathcal{N}, \mathcal{A}, u)$, where:

• $\mathcal{N}={1, \ldots, N}$ is a finite set of $N$ players, indexed by $i$.
• $A=A_1 \times \cdots \times A_N$ is a set of jnint actions of the players, where $\mathcal{A}_i$ is a finite set of actions available to player $i . a=\left(a_1, \ldots, a_N\right) \in \mathcal{A}$ is called an action profile with $a_i \in \mathcal{A}_i$.
• $u=\left(u_1, \ldots, u_N\right)$ where $u_i: \mathcal{A} \mapsto \mathbb{R}$ is a utility (or payoff) function for player $i$. It maps an action profile $a$ to a real value. An important characteristic of a game is that player $i$ ‘s utility depends on not only his own action $a_i$ but also the actions taken by other players; thus, the utility function is defined over the space of $\mathcal{A}$ instead of $\mathcal{A}i$. A player can choose a mixed strategy. We use $S_i=\Delta^{\left|\mathcal{A}_l\right|}$ to denote the set of mixed strategies for player $i$, which is the probability simplex with dimension $\left|\mathcal{A}_i\right|$. Similarly, $S=S_1 \times \cdots \times S_N$ is the set of joint strategies and $s=\left(s_1, \ldots, s_N\right) \in S$ is called a strategy profile. The support of a mixed strategy is defined as the set of actions that are chosen with a nonzero probability. An action of player $i$ is a pure strategy and can be represented by a probability distribution with support size 1 (of value 1 in one dimension and 0 in all other dimensions). The utility function can be extended to mixed strategies by using expected utility. That is, if we use $s_l\left(a_i\right)$ to represent the probability of choosing action $a_i$ in strategy $s_i$, the expected utility for player $i$ given strategy profile $s$ is $u_i(s)=\sum{a \in \mathcal{A}} u_i(a) \prod_{l^{\prime}=1}^n s_{l^{\prime}}\left(a_{l^{\prime}}\right)$. A game is zero-sum if the utilities of all the players always sum up to zero, i.e. $\sum_l u_i(s)=0, \forall s$ and is nonzero-sum or general-sum otherwise.
Many classic games can be represented in normal form. Table 2.3 shows the game Prisoner’s Dilemma (PD). Each player can choose between two actions, Cooperate (C) and Defect (D). If they both choose $\mathrm{C}$, they both suffer a small loss of -1 . If they both choose $\mathrm{D}$, they both suffer a big loss of -2 . However, if one chooses $\mathrm{C}$ and the other chooses $\mathrm{D}$, the one who chooses $\mathrm{C}$ suffers a huge loss while the other one does not suffer any loss. If the row player chooses a mixed strategy of playing C with probability 0.4 and D 0.6 , while the column player chooses the uniform random strategy, then the row player’s expected utility is $-1.4=(-1) \cdot 0.4 \cdot 0.5+(-3) \cdot 0.4 \cdot 0.5+(-2) \cdot 0.6 \cdot 0.5$. Now we provide the formal definition of best response. We use $-i$ to denote all players but $i$.

## 经济代写|博弈论代写Game Theory代考|Extensive-Form Games

Extensive-form games (EFGs) represent the sequential interaction of players using a rooted game tree. Figure 2.1 shows a simple example game tree. Each node in the tree belongs to one of the players and corresponds to a decision point for that player. Outgoing edges from a node represent actions that the corresponding player can take. The game starts from the root node, with the player corresponding to the root node taking an action first. The chosen action brings the game to the child node, and the corresponding player at the child node takes an action. The game continues until it reaches a leaf node (also called a terminal node), i.e. each leaf node in the game tree is a possible end state of the game. Each leaf node is associated with a tuple of utilities or payoffs that the players will receive when the game ends in that state. In the example in Figure 2.1, there are three nodes. Node 1 belongs to Player 1 (P1), and nodes 2 and 3 belong to Player 2 (P2). Player 1 first chooses between action L and R, and then Player 2 chooses between action 1 and r. Player 1 ‘s highest utility is achieved when Player 1 chooses L, and Player 2 chooses 1 .

There is sometimes a special fictitious player called Chance (or Nature), who takes an action according to a predefined probability distribution. This player represents the stochasticity in many problems. For example, in the game of Poker, each player gets a few cards that are randomly dealt. This can be represented by having a Chance player taking an action of dealing cards. Unlike the other real players, the Chance player does not rationally choose an action to maximize his utility since he does not have a utility function. In the special case where Nature only takes an action at the very beginning of the game, i.e. the root of the game tree, the game is essentially a Bayesian game, as we will detail in Section 2.7.

In a perfect information game, every player can perfectly observe the action taken by players in the previous decision points. For example, when two players are playing the classic board game of Go or Tic-Tac-Toe, each player can observe the other players’ previous moves before she decides her move. However, it is not the case in many other problems. An EFG can also capture imperfect information, i.e. a game where players are sometimes uncertain about the actions taken by other players and thus do not know which node they are at exactly when they take actions. The set of nodes belonging to each player is partitioned into several information sets. The nodes in the same information set cannot be distinguished by the player that owns those nodes. In other words, the player knows that she is at one of the nodes that belong to the same information set, but does not know which one exactly. For example, in a game of Poker where each player has private cards, a player cannot distinguish between certain nodes that only differ in the other players’ private cards. Nodes in the same information set must have the same set of actions since otherwise, a player can distinguish them by checking the action set. It is possible that an information set only contains one node, i.e. a singleton. If all information sets are singletons, the game is a perfect information game. The strategy of a player specifies what action to take at each information set. In the example game in Figure 2.1, the dashed box indicates that nodes 2 and 3 are in the same information set, and Player 2 cannot distinguish between them. Thus, nodes 2 and 3 have the same action set. This information set effectively makes the example game a simultaneous game as Player 2 has no information about Player 1s previous actions when he makes a move.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|Normal-Form Games

• $\mathcal{N}=1, \ldots, N$ 是一个有限集 $N$ 球员，索引i.
• $A=A_1 \times \cdots \times A_N$ 是玩家的一组 jnint 动作，其中 $\mathcal{A}_i$ 是玩家可用的一组有限动作 i. $a=\left(a_1, \ldots, a_N\right) \in \mathcal{A}$ 被称为动作配置文件 $a_i \in \mathcal{A}_i$.
• $u=\left(u_1, \ldots, u_N\right)$ 在哪里 $u_i: \mathcal{A} \mapsto \mathbb{R}$ 是玩家的效用 (或收益) 函数 $i$. 它映射了一个动作配置文 件 $a$ 到一个真正的价值。游戏的一个重要特征是玩家 $i$ 的效用不仅取决于他自己的行动 $a_i$ 还有其他玩 家采取的行动；因此，效用函数定义在 $\mathcal{A}$ 代替 $\mathcal{A} i$. 玩家可以选择混合策略。我们用 $S_i=\Delta^{\left|\mathcal{A}l\right|}$ 表 示玩家的混合策略集 $i$ ，这是具有维度的概率单纯形 $\left|\mathcal{A}_i\right|$. 相似地， $S=S_1 \times \cdots \times S_N$ 是一组联 合策略，并且 $s=\left(s_1, \ldots, s_N\right) \in S$ 称为策略配置文件。混合策略的支持被定义为以非零概率选 择的一组动作。玩家的一个动作 $i$ 是一种纯策略，可以用支持大小为 1 的概率分布表示（在一个维 度上值为 1 ，在所有其他维度上值为 0 ) 。效用函数可以通过使用期望效用扩展到混合策略。也就 是说，如果我们使用 $s_l\left(a_i\right)$ 表示选择动作的概率 $a_i$ 在战略上 $s_i$ ，玩家的预期效用 $i$ 给定的战略概兄 $s$ 是 $u_i(s)=\sum a \in \mathcal{A} u_i(a) \prod{l^{\prime}=1}^n s_{l^{\prime}}\left(a_{l^{\prime}}\right)$. 如果所有玩家的效用总和为零，则游戏是零和游戏， 即 $\sum_l u_i(s)=0, \forall s$ 否则是非零和或一般和。
许多经典游戏都可以用正常形式表示。表 2.3 显示了囚徒困境 (PD) 游戏。每个玩家都可以在合作 (C) 和背叛 (D) 两种行动之间进行选择。如果他们都选择C，他们都遭受 -1 的小损失。如果他们都选 择 $\mathrm{D}$ ，他们都遭受了 -2 的巨大损失。然而，如果选择C另一个选择D，选择的人C遭受巨大损失， 而另一个则没有遭受任何损失。如果行玩家选择以概率 0.4 和 D 0.6 玩 C 的混合策略，而列玩家选 择均匀随机策略，则行玩家的期望效用为
$-1.4=(-1) \cdot 0.4 \cdot 0.5+(-3) \cdot 0.4 \cdot 0.5+(-2) \cdot 0.6 \cdot 0.5$. 现在我们提供最佳响应的正式 定义。我们用 $-i$ 表示所有玩家，但 $i$.

## 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|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

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

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

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

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

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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。