### 经济代写|博弈论代写Game Theory代考|ECON2070

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Bayesian updating with relative quality

The qualities, $q^{\prime}$ and $q$, of two randomly selected contestants are independent and normally distributed with mean $\mu_{q}$ and variance $\sigma_{q}^{2}$. The difference in quality, $h=q^{\prime}-q$ is normally distributed with mean 0 and variance $2 \sigma_{q}^{2}$. This is the prior distribution of quality difference before any observation. On obtaining the observation $\xi^{\prime}$ the $q^{\prime}$ contestant updates this prior to a posterior distribution using Bayes’ theorem. This theorem shows that the posterior density function for $h$ is proportional to the product
prior density of $h \times$ conditional density of $\xi^{\prime}$ given $h$.
From eq (3.22), the observation $\xi^{\prime}$ given $h$ is normally distributed with mean $h$ and variance $\sigma^{2}$. After some manipulation, using eq (3.24), one finds that the posterior distribution of $h$ given $\xi^{\prime}$ is normal with mean $\kappa \xi^{\prime}$ and variance $\kappa \sigma^{2}$, where $\kappa=2 \sigma_{q}^{2} /\left(2 \sigma_{q}^{2}+\sigma^{2}\right)$.

Now let $x$ be the resident threshold strategy of using action $\mathrm{A}$ if $\xi>x$. For a mutantresident pair with difference $h$, the probability that the resident individual uses $\mathrm{A}$ is then $p_{A}(h ; x)=\mathrm{P}(\epsilon>x+h)$. The payoffs for choosing action $\mathrm{A}$ and $\mathrm{S}$, given a true quality difference $h$ is then
\begin{aligned} &w_{A}(h ; x)=\left(1-p_{A}(h ; x)\right) V-p_{A}(h ; x) C e^{-h} \ &w_{S}(h ; x)=\left(1-p_{A}(h ; x)\right) \frac{V}{2} \end{aligned}
where we used eq (3.23) for the cost. The mutant individual of course does not know $h$. Instead, using the posterior distribution, the mutant payoffs for $\mathrm{A}$ and $\mathrm{S}$ given the observation $\xi^{\prime}$ are
\begin{aligned} &W_{A}\left(\xi^{\prime} ; x\right)=\int_{-\infty}^{\infty} w_{A}(h ; x) f\left(h \mid \xi^{\prime}\right) d h \ &W_{S}\left(\xi^{\prime} ; x\right)=\int_{-\infty}^{\infty} w_{S}(h ; x) f\left(h \mid \xi^{\prime}\right) d h \end{aligned}
where $f\left(h \mid \xi^{\prime}\right)$ is the posterior probability density function.

## 经济代写|博弈论代写Game Theory代考|Stability Concepts: Beyond Nash Equilibria

Gene frequencies in a population change over time as result of natural selection and random processes. So far we have not specified the details of these evolutionary dynamics. Nevertheless, in Chapter 1 we argued that if the dynamics have a stable endpoint, then at this endpoint no mutant strategy should outperform the resident strategy so that the resident strategy is a Nash equilibrium; i.e. condition (2.1) holds for invasion fitness $\lambda$, or equivalently condition (2.4) holds for a fitness proxy $W$. However, we have yet to deal with two central issues, which form the main focus of this chapter:

• Stability against invasion by mutants. The Nash condition is necessary for stability but is it sufficient? For example, in the Hawk-Dove game with $V<C$, at the Nash equilibrium every mutant strategy does equally well as the resident. So can mutant numbers increase by random drift, changing the population composition? In considering conditions that are sufficient to ensure stability we will assume that mutants arise one at a time and their fate is determined before any other mutant arises. As we describe, this leads to the concept of an Evolutionarily Stable Strategy $(\mathrm{ESS})$
• Dynamic stability and attainability. Even if a Nash equilibrium cannot be invaded by new mutants, if the initial population is not at the equilibrium, will the evolutionary process take the population to it? To consider this question we introduce the idca of convergence stability. As we describe, a strategy $x^{}$ is convergence stable if the resident strategy evolves to this strategy provided that the initial resident strategy is sufficiently close to $x^{}$. In other words, if the whole population is perturbed away from $x^{}$ then it will evolve back to $x^{}$.

## 经济代写|博弈论代写Game Theory代考|Bayesian updating with relative quality

$$w_{A}(h ; x)=\left(1-p_{A}(h ; x)\right) V-p_{A}(h ; x) C e^{-h} \quad w_{S}(h ; x)=\left(1-p_{A}(h ; x)\right) \frac{V}{2}$$

$$W_{A}\left(\xi^{\prime} ; x\right)=\int_{-\infty}^{\infty} w_{A}(h ; x) f\left(h \mid \xi^{\prime}\right) d h \quad W_{S}\left(\xi^{\prime} ; x\right)=\int_{-\infty}^{\infty} w_{S}(h ; x) f\left(h \mid \xi^{\prime}\right) d h$$

## 经济代写|博弈论代写Game Theory代考|Stability Concepts: Beyond Nash Equilibria

• 抵抗突变体入侵的稳定性。纳什条件对于稳定性是必要的，但它是否充分？例如，在鹰鸽游戏中在<C，在纳什均衡下，每个突变策略都与常驻策略一样好。那么突变数量可以通过随机漂移增加，改变种群组成吗？在考虑足以确保稳定性的条件时，我们将假设突变体一次出现一个，并且它们的命运在任何其他突变体出现之前就已确定。正如我们所描述的，这导致了进化稳定策略的概念(和小号小号)
• 动态稳定性和可达性。即使一个纳什均衡不能被新的突变体入侵，如果初始种群不处于平衡状态，进化过程是否会将种群带向它？为了考虑这个问题，我们引入了收敛稳定性的 idca。正如我们所描述的，一种策略X如果驻留策略演变为该策略，则收敛稳定，前提是初始驻留策略足够接近X. 换句话说，如果整个人口都被扰乱X然后它会进化回X.

## 有限元方法代写

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## MATLAB代写

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