Posted on Categories:Microeconomics, 微观经济学, 经济代写

# 经济代写|微观经济学代考Microeconomics代写|Population dynamics

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## 经济代写|微观经济学代考Microeconomics代写|Evolutionary stability

The notion of evolutionary stability was first defined, without reference to an explicit dynamic process, and in the context of symmetric games. One considers a population of individuals that choose pure strategies in $X$. If $p$ is a mixed strategy, for any pure strategy $x \in X$, the number $p(x)$ can be interpreted as the proportion in the population of individuals that use the strategy $x$. The expected utility $u(x, p)=\sum_{y \in X} u(x, y) p(y)$ is the utility obtained on average by the use of $x$, in uniform interactions with a population characterized by $p$. Let $p$ and $q$ be two mixed strategies, and consider the mix of two sub-populations respectively characterized by $p$ and $q$, in proportions $(1-\varepsilon)$ and $\varepsilon$. In the whole population, individuals of the first sub-population get on average the utility $u(p,(1-\varepsilon) p+\varepsilon q)$ and the other individuals the utility $u(q,(1-\varepsilon) p+\varepsilon q)$. Population dynamics will make the assumption that, in such a situation, the ratio $(1-\varepsilon) / \varepsilon$ will be modified in favor of the sub-population that gets, on average, the largest utility (see next section).

By definition, a mixed strategy $p$ is evolutionary stable if, for any other mixed strategy $q \neq p$, there exists an “invasion barrier” $\varepsilon>0$ such that:
$$\forall \varepsilon \in] 0, \bar{\varepsilon} \mid, u(p,(1-\varepsilon) p+\varepsilon q)>u(q,(1-\varepsilon) p+\varepsilon q)$$
This means that a population that plays the mixed strategy $p$ cannot be “invaded” by any deviant small sub-population. Indeed, if such a small sub-population were to appear, its size would immediatly decrease even further.

## 经济代写|微观经济学代考Microeconomics代写|Replicator dynamics

An evolutionary dynamics partially specifies the intuitions that were evoked by the preceding definitions of evolutionary stability. The use of a strategy tends to increase if this strategy performed relatively well in the past. The canonical example here is the “replicator dynamics”, which appears as a central concept in many selection processes. There exist several variants of that dynamics, defined at the aggregate level, leaving unspecified the matching process between individuals. The common principle is that the proportion of individuals who use a given strategy changes over time; the speed of this change being proportional to the difference between the average utility obtained by the users of this strategy and the average utility in the whole population, at the current date. Replicator dynamics is a model of how the best performing strategies within a population are selected. Whenever a strategy is not present at all in the population, no mutation can make it appear in the future.

Consider first a single population, and thus a symmetric game. Let $p^t$ be the vector that describes, at date $t$, the proportion, among the population, of users of the different pure strategies. The expected utility to the pure strategy $x$ is $u\left(x, p^t\right)$, interpreted as the average payoff to the users of strategy $x$ when they are randomly matched with any individual. The average utility in the population is $u\left(p^t, p^t\right)$. One supposes that, in continuous time, the speed of variation $p^t(x)$ is exactly proportional to $u\left(x, p^t\right)-u\left(p^t, p^t\right)$. One then obtains the one-population replication equation:
$$p^t(x)=v p^t(x)\left[u\left(x, p^t\right)-u\left(p^t, p^t\right)\right]$$
with $v$ a speed coefficient, which can depend upon time, but is the same for the different strategies (one usually takes $v=1$ ). This equation has a dominant interpretation in terms of biological selection, but can also be interpreted in terms of social imitation, as is proven in the third sub-section. In the two-population case, several generalisations of the replication equation make sense. One can for instance propose the “standard twopopulation replicator dynamics”:
$$\begin{gathered} \left(\left(p_1^t\left(x_1\right)=p_1^t\left(x_1\right)\left[u_1\left(x_1, p_2^t\right)-u_1\left(p_1^t, p_2^t\right)\right]\right.\right. \ \left(\left(p_2^t\left(x_2\right)=p_2^t\left(x_2\right)\left[u_2\left(p_1^t, x_2\right)-u_2\left(p_1^t, p_2^t\right)\right]\right.\right. \end{gathered}$$

# 微观经济学代写

## 经济代写|微观经济学代考Microeconomics代写|Evolutionary stability

$$\forall \varepsilon \in] 0, \bar{\varepsilon} \mid, u(p,(1-\varepsilon) p+\varepsilon q)>u(q,(1-\varepsilon) p+\varepsilon q)$$

## 经济代写|微观经济学代考Microeconomics代写|Replicator dynamics

$$p^t(x)=v p^t(x)\left[u\left(x, p^t\right)-u\left(p^t, p^t\right)\right]$$

$$\left(\left(p_1^t\left(x_1\right)=p_1^t\left(x_1\right)\left[u_1\left(x_1, p_2^t\right)-u_1\left(p_1^t, p_2^t\right)\right] \quad\left(\left(p_2^t\left(x_2\right)=p_2^t\left(x_2\right)\left[u_2\left(p_1^t, x_2\right)-u_2\left(p_1^t, p_2^t\right)\right]\right.\right.\right.\right.$$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。