## 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|Fair odds

An investment into an opportunity $A$ offering a return of $r \geq 1$ euros per euro invested with a certain probability $\operatorname{Pr}(A)$ or returning nothing (with probability $1-\operatorname{Pr}(A)$ ) is called a bet on $A$. The investor is then a bettor (or a gambler) and the net return
$$\rho=r-1$$
is the payoff of the bet. The payoff is assumed to be guaranteed by a bookmaker (or bank). The net return rate is also denoted ${ }^3$ by $\rho: 1$ and known as the odds of the bet.

The expected gain (per euro) of the gambler, and hence the bookmaker’s loss, is
$$E=\rho \operatorname{Pr}(A)+(-1)(1-\operatorname{Pr}(A))=r \operatorname{Pr}(A)-1$$

The odds $\rho: 1$ are considered to be fair if the gambler and the bookmaker have the same expected gain, i.e., if
$$E=-E \quad \text { and hence } \quad E=0$$
holds. In other words:
$\rho: 1 \quad$ is fair $\Longleftrightarrow \rho=\frac{1-\operatorname{Pr}(A)}{\operatorname{Pr}(A)} \Longleftrightarrow r=\frac{1}{\operatorname{Pr}(A)}$
If the true probability $\operatorname{Pr}(A)$ is not known to the bettor, it needs to be estimated. Suppose the bettor’s estimate for $\operatorname{Pr}(A)$ is $p$. Then the bet appears (subjectively) advantageous if and only if
$$E(p)>0, \quad \text { i.e., if } r>1 / p .$$
The bettor will consider the odds $\rho: 1$ as fair if
$$E(p)=0 \text { and hence } r=1 / p \text {. }$$
In the case $E(p)<0$, of course, the bettor would not expect a gain but a loss on the bet – on the basis of the information that has led to the subjective probability estimate $p$ for $\operatorname{Pr}(A)$.

## 经济代写|博弈论代写Game Theory代考|Betting on alternatives

Consider $k$ mutually exclusive events $A_0, A_1, \ldots, A_{k-1}$ of which one will occur with certainty and a bank that offers the odds $\rho_i: 1$ for bets on the $k$ events $A_i$, which means:
(1) The bank guarantees a total payoff of $r_i=\rho_i+1$ euros for each euro invested in $A_i$ if the event $A_i$ occurs.
(2) The bank offers a scenario with $1 / r_i$ being the probability for $A_i$ to occur.
Suppose a gambler estimates the events $A_i$ to occur with probabilities $p_i$ and decides to invest the capital $B$ of unit size ${ }^6$ $b=1$ fully. Under this condition, a (betting) strategy is a $k$-tuple $a=\left(a_0, a_1, \ldots, a_{k-1}\right)$ of numbers $a_i \geq 0$ such that
$$a_0+a_1+\ldots+a_{k-1}=1$$
with the interpretation that the portion $a_i$ of the capital will be bet onto the occurrence of event $A_i$ for $i=0,1, \ldots, k-1$. The gambler’s expected logarithmic utility of strategy $a$ is
\begin{aligned} U(a, p) &=\sum_{i=0}^{k-1} p_i \ln \left(a_i r_i\right) \ &=\sum_{i=0}^{k-1} p_i \ln a_i+\sum_{i=0}^{k-1} p_i \ln r_i . \end{aligned}
Notice that $p=\left(p_0, p_1, \ldots, p_{k-1}\right)$ is a strategy in its own right and that the second sum term in the expression for $U(a, p)$ does not depend on the choice of $a$. So only the first sum term is of interest when the gambler seeks a strategy with optimal expected utility.

## 经济代写|博弈论代写博弈论代考|公平赔率

$$\rho=r-1$$

$$E=\rho \operatorname{Pr}(A)+(-1)(1-\operatorname{Pr}(A))=r \operatorname{Pr}(A)-1$$

$$E=-E \quad \text { and hence } \quad E=0$$

$\rho: 1 \quad$是公平的$\Longleftrightarrow \rho=\frac{1-\operatorname{Pr}(A)}{\operatorname{Pr}(A)} \Longleftrightarrow r=\frac{1}{\operatorname{Pr}(A)}$

$$E(p)>0, \quad \text { i.e., if } r>1 / p .$$

$$E(p)=0 \text { and hence } r=1 / p \text {. }$$

## 经济代写|博弈论代写博弈论代考|在替代品上下注

(1)银行保证支付总额为 $r_i=\rho_i+1$ 每投资一欧元 $A_i$ 如果事件 $A_i$
(2)银行提供了一个场景 $1 / r_i$ 是概率 $A_i$ 发生:发生假设一个赌徒估计事件 $A_i$ 以概率发生 $p_i$ 并决定进行资本投资 $B$ 单位尺寸 ${ }^6$ $b=1$ 完全。在这种情况下，一个(赌博)策略是一个 $k$-tuple $a=\left(a_0, a_1, \ldots, a_{k-1}\right)$ 关于数字 $a_i \geq 0$ 这样
$$a_0+a_1+\ldots+a_{k-1}=1$$

\begin{aligned} U(a, p) &=\sum_{i=0}^{k-1} p_i \ln \left(a_i r_i\right) \ &=\sum_{i=0}^{k-1} p_i \ln a_i+\sum_{i=0}^{k-1} p_i \ln r_i . \end{aligned}
$p=\left(p_0, p_1, \ldots, p_{k-1}\right)$ 策略是否有它自己的权利，表达式中的第二个和项 $U(a, p)$ 不就看选择了吗 $a$。因此，当赌徒寻求具有最佳期望效用的策略时，只有第一项和是有意义的

## 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|Expected utility

With respect to the logarithmic utility function $\ln x$, the expected utility of an investment of size $x$ with net return rate $\rho$ would be
$$U(x)=p \ln (b+\rho x))+q \ln (b-x) .$$
The marginal utility of $x$ is the value of the derivative
$$U^{\prime}(x)=\frac{\rho p}{b+\rho x}-\frac{q}{b-x} .$$
The second derivative is
$$U^{\prime \prime}(x)=\frac{\rho^2 p}{(b-\rho x)^2}+\frac{q}{(b-x)^2}>0 .$$
So the investment $x^$ has optimal (logarithmic) utility if $$U^{\prime}\left(x^\right)=0 \quad \text { or } \quad \frac{\rho p}{b+\rho x^}=\frac{q}{b-x^} .$$
Ex. 4.4. In the situation of Ex. 4.3, one has
$$U(x)=\frac{1}{10} \ln (b+99 x)+\frac{9}{10} \ln (b-x)$$
with the derivative
$$U^{\prime}(x)=\frac{99}{10(b+99 x)}-\frac{9}{10(b-x)}$$
$U^{\prime}\left(x^\right)=0$ implies $x^=b / 11$. Hence the portion $a^*=1 / 11$ of the portfolio should be invested in order to maximize the expected utility $U$. The rest of the portfolio should be retained and not be invested.

## 经济代写|博弈论代写Game Theory代考|The fortune formula

Let $a=x / b$ denote the fraction of the portfolio $B$ to be possibly invested into the opportunity $A$ with $r$-fold return if $A$ works out. With $\rho=r-1>0$, the expected logarithmic utility function then becomes
$$u(a)=U(x / b)=p \ln b(1+\rho x / b)+q \ln b(1-x / b)$$
with the derivative
$$u^{\prime}(a)=\frac{\rho p}{1+\rho a}-\frac{q}{1-a}$$
If a loss is to be expected with positive probability $q>0$, and the investor decides on a full investment, i.e., chooses $a=1$, then the expected utility (value)
$$u(1)=\lim _{a \rightarrow 0} u(a)=-\infty$$
results – no matter how big the net return rate $\rho$ might be.
On the other hand, the choice $a=0$ of no investment has the utility
$$u(0)=\ln b .$$
The investment fraction $a^*$ yielding the optimal utility lies somewhere between these extremes.

## 经济代写|博弈论代写博弈论代考|期望效用

$$U(x)=p \ln (b+\rho x))+q \ln (b-x) .$$

$$U^{\prime}(x)=\frac{\rho p}{b+\rho x}-\frac{q}{b-x} .$$二阶导数为
$$U^{\prime \prime}(x)=\frac{\rho^2 p}{(b-\rho x)^2}+\frac{q}{(b-x)^2}>0 .$$所以投资 $x^$ 有最佳(对数)效用，如果 $$U^{\prime}\left(x^\right)=0 \quad \text { or } \quad \frac{\rho p}{b+\rho x^}=\frac{q}{b-x^} .$$
Ex;4.4. 在例4.3的情况下，有
$$U(x)=\frac{1}{10} \ln (b+99 x)+\frac{9}{10} \ln (b-x)$$
，导数
$$U^{\prime}(x)=\frac{99}{10(b+99 x)}-\frac{9}{10(b-x)}$$
$U^{\prime}\left(x^\right)=0$ 暗示 $x^=b / 11$。因此这部分 $a^*=1 / 11$ 的投资组合应该是为了最大化预期效用 $U$。投资组合的其余部分应该被保留，而不是用于投资

## 经济代写|博弈论代写博弈论代考|财富公式

$$u(a)=U(x / b)=p \ln b(1+\rho x / b)+q \ln b(1-x / b)$$
，导数
$$u^{\prime}(a)=\frac{\rho p}{1+\rho a}-\frac{q}{1-a}$$

$$u(1)=\lim _{a \rightarrow 0} u(a)=-\infty$$

$$u(0)=\ln b .$$

## 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|Investing and Betting

Assume that an investor (or bettor or gambler or simply player) is considering a financial engagement in a certain venture. Then the obvious – albeit rather vague – big question for the investor is:

• What decision should best be taken?
More specifically, the investor wants to decide whether an engagement is worthwhile at all and, if so, how much of the available capital should be invested how. Obviously, the answer depends on additional information: What is the likelihood of a success? What gain can be expected? What is the risk of a loss? etc.

The investor is thus about to participate as a player in a 2-person game with an opponent whose strategies and objective are not always clear or known in advance. Relevant information is not completely (or not reliably) available to the investor so that the decision must be made under uncertainties. Typical examples are gambling and betting where the success of the engagement depends on events that may or may not occur and hence on “fortune” or “chance”. But also investments in the stock market fall into this category when it is not clear in advance whether the value of a particular investment will rise or fall.

We are not able to answer the big question above completely but will discuss various aspects of it. Before going into further details, let us illustrate the difficulties of the subject with a classical – and seemingly paradoxical – gambling situation.

The St. Petersburg paradox. Imagine yourself as a potential player in the following game of chance.

## 经济代写|博弈论代写Game Theory代考|Proportional investing

Our general model consists of a potential investor with an initial portfolio $B$ of $b>0$ euros (or dollars or…) and an investment opportunity $A$. If things go well, an investment of size $x$ would bring a return $r x>x$. If things do not go well, the investment will return nothing.
In the analysis, we will denote the net return rate by
$$\rho=r-1$$
The investor is to decide what portion of $B$ should be invested. The investor believes:
(PI) Things go well with probability $p>0$ and do not go well with probability $q=1-p$.

Under the assumption (PI), the investor’s expected portfolio value after the investment $x$ is
$$B(x)=[(b-x)+r x] p+(b-x) q=[b+\rho x] p+(b-x) q$$
since an amount of size $b-x$ is not invested and therefore not at risk. The derivative is
$$B^{\prime}(x)=\rho p-q$$
So $B(x)$ is strictly increasing if $\rho>q / p$ and non-increasing otherwise. Hence, if the investor’s decision is motivated by the maximization of the expected portfolio value $B(x)$, the naive investment rule applies:
(NIR) If $\rho>q / p$, invest all of $B$ in $A$ and expect the return $B(b)=r b p=(1+q) b>b$.
If $\rho \leq q / p$, invest nothing since no proper gain is expected.
In spite of its intuitive appeal, rule (NIR) can be quite risky (see Ex. 4.3).

## 经济代写|博弈论代写博弈论代考|比例投资

A variety of game-theoretic models dealing with a range of phenomena have previously been developed. In Chapter 3 we present many of what have become the standard models. Some of these standard models deal with cooperation and the contribution to a common good, including parental care. We also introduce the simplest model of animal conflict over a resource: the Hawk-Dove game. Many animals signal to others, and we present a simple model showing that signals can evolve from cues, later returning to the question of why signals should be honest in Section 7.4. We also present the standard model of sex allocation, a topic we later return to in Section 10.4. Most of these simple models assume that all individuals are the same, so that if they take different actions this is because their choice has a random component. In reality it is likely that individuals differ in aspects of their state such as size or fighting ability, and different behaviours are a result of these difference. At the end of Chapter 3 we illustrate how such state-dependent decision making can be incorporated into models. The effects of state differences are harder to analyse when the state of offspring is affected by the state and action(s) of their parent(s). We defer the description of some standard models that have these features until Chapter 10 , where we outline the theory necded to analysc inter-gencrational cffects. Wc apply this theory to the problem of sex allocation when offspring tend to inherit the quality of their mother (Trivers and Willard theory) and to the case where female preference for a male ornament (such as tail length) and the ornament co-evolve (the Fisher process).

## 经济代写|博弈论代写Game Theory代考|Challenges

The approach taken in the standard game-theoretic models often rests on idealized assumptions. This is important and helpful in providing easily understandable and clcar predictions, but biologists might rcly on modcls without carcful cxamination of the consequences of the assumptions and limitations of the models. We believe the ideas used in the field need to be re-evaluated and updated. In particular, game theory needs to be richer, and much of the remainder of the book is concerned with ways in which we believe it should be enriched. Here we outline some of these ways.

Co-evolution. There is a tendency to consider the evolution of a single trait keeping other traits fixed. It is often the case, however, that another trait strongly interacts with the focal trait (Chapter 6). Co-evolution of the two traits can for instance bring about disruptive selection causing two morphs to coexist or giving rise to two evolutionarily stable outcomes. These insights might not be gained if the traits are considered singly. Variation. There is usually considerable variation in natural populations. Many existing game-theoretical models ignore this and assume all individuals in a given role are the same. However, variation affects the degree to which individuals should be choosy over who they interact with and the value of expending effort observing others. Variation thus leads to co-evolution of the focal trait with either choosiness or social sensitivity. These issues are crucial for effects of reputation and the functioning of biological markets (Chapter 7). Variation is also crucial for phenomena such as signalling. We believe that models need to be explicit about the type and amount of variation present, and to explore the consequences.

Process. In many game-theoretical models of the interaction of two individuals each chooses its action without knowledge of the choice of their partner. Furthermore, neither alters its choice once the action of its partner has been revealed (a simultaneous or one-shot game). In reality most interactions involve individuals responding to each other. The final outcome of the interaction can then be thought of as resulting from some interaction process. The outcome can depend strongly on the nature of this process (Chapter 8). Since partners vary, the interaction often involves gaining information about the abilities and intentions of other individuals, emphasizing the importance of variation and learning (Chapters 5 and 8).

## 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。