标签： ECON2070

经济代写|博弈论代写Game Theory代考|PSCI288

Posted on 2023年8月14日2023年8月28日 by statistics-lab

如果你也在怎样代写博弈论Game theory 这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。博弈论Game theory在20世纪50年代被许多学者广泛地发展。它在20世纪70年代被明确地应用于进化论，尽管类似的发展至少可以追溯到20世纪30年代。博弈论已被广泛认为是许多领域的重要工具。截至2020年，随着诺贝尔经济学纪念奖被授予博弈理论家保罗-米尔格伦和罗伯特-B-威尔逊，已有15位博弈理论家获得了诺贝尔经济学奖。约翰-梅纳德-史密斯因其对进化博弈论的应用而被授予克拉福德奖。

博弈论Game theory是对理性主体之间战略互动的数学模型的研究。它在社会科学的所有领域，以及逻辑学、系统科学和计算机科学中都有应用。最初，它针对的是两人的零和博弈，其中每个参与者的收益或损失都与其他参与者的收益或损失完全平衡。在21世纪，博弈论适用于广泛的行为关系；它现在是人类、动物以及计算机的逻辑决策科学的一个总称。

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。

经济代写|博弈论代写Game Theory代考|Mechanism Design and the Revelation Principle

This section develops the general version of the mechanism-design problem and shows how it can be simplified using the revelation principle.

We suppose that there are $I+1$ players: a principal (player 0 ) with no private information, and $I$ agents $(i=1, \ldots, I)$ with types $\theta=\left(\theta_1, \ldots, \theta_1\right)$ in some set $\Theta$. For the time being, we can allow the probability distribution on $\Theta$ to be quite general, requiring only that expectations and conditional expectations of the utility functions be well defined.

The object of the mechanism built by the principal is to determine an allecation $y={x, t}$. An allocation consists of a vector $x$, called a decision, belonging to a compact, convex, nonempty $\mathscr{X} \subset \mathbb{R}^n$, and a vector of monetary transfers $t=\left(t_1, \ldots, t_I\right)$ from the principal to each agent (which can he positive or negative). ${ }^8$ In most applications $\mathscr{X}$ is taken large enough that we are ensured an interior solution; one exception is the auction example mentioned above.

Player $i(i=0,1, \ldots, I)$ has a von Neumann-Morgenstern utility $u_i(y, \theta)$. We will assume that $u_i(i=1, \ldots, I)$ is strictly increasing in $t_i$, that $u_0$ is decreasing in each $t_i$, and that these functions are twice continuously differentiable.

Given a (type-contingent) allocation ${y(\theta)}_{\theta \in \Theta}$, agent $i(i=1, \ldots, I)$ with type $\theta_i$ has expected or “interim” utility
$$
U_i\left(\theta_i\right) \equiv \mathrm{F}\theta{ }_i\left[u_i\left(y\left(\theta_i, \theta{-i}\right), \theta_i, \theta_{-i}\right) \mid \theta_i\right]
$$
and the principal has expected utility
$$
\mathrm{I}_{i 0} u_0(y(\theta), \theta) .
$$

经济代写|博弈论代写Game Theory代考|Mechanism Design with a Single Agent

The following methodology, first developed by Mirrlees (1971), was extended and applied to various contexts by Mussa and Rosen (1978), Baron and Myerson (1982), and Maskin and Riley (1984a), among others. The presentation, including the propositions, follows the general analysis of Guesnerie and Laffont (1984). ${ }^{11}$

Because there is a single agent, we omit the subscripts on transfer $(t)$ and type $(\theta)$ in this section. We assume that the agent’s type lies in an interval $[\theta, \theta]$. The agent knows $\theta$, and the principal has the prior cumulative distribution function $P(P(\theta)=0, P(\theta)=1)$, with differentiable density $p(\theta)$ such that $p(\theta)>0$ for all $\theta$ in $[\theta, \bar{\theta}]$. (Differentiability of the density is not necessary, but is assumed for convenience.) The type space is single dimensional, ${ }^{12}$ but the decision space may be multidimensional. (Although we consider a multidimensional decision for completeness, the reader can grasp the main ideas from the case of a single-dimensional decision.) A (type-contingent) allocation is a function from the agent’s type into an allocation:
$$
\theta \rightarrow y(\theta)=(x(\theta), t(\theta)) .
$$

博弈论代考

经济代写|博弈论代写Game Theory代考|Mechanism Design and the Revelation Principle

本节发展了机制设计问题的一般版本，并展示了如何使用启示原则将其简化。

我们假设有$I+1$个玩家:一个没有私有信息的主体(玩家0)，以及在某个集合$\Theta$中类型为$\theta=\left(\theta_1, \ldots, \theta_1\right)$的$I$个代理$(i=1, \ldots, I)$。目前，我们可以允许$\Theta$上的概率分布相当一般，只需要很好地定义效用函数的期望和条件期望。

主体构建的机制的目标是确定一个断言$y={x, t}$。分配由一个矢量$x$(称为决策)组成，它属于一个紧凑的、凸的、非空的$\mathscr{X} \subset \mathbb{R}^n$，以及一个从委托人到每个代理人的货币转移矢量$t=\left(t_1, \ldots, t_I\right)$(可以是正的，也可以是负的)。${ }^8$在大多数应用中，$\mathscr{X}$足够大，我们可以确保内部解决方案;一个例外是上面提到的拍卖例子。

玩家$i(i=0,1, \ldots, I)$有一个von Neumann-Morgenstern实用程序$u_i(y, \theta)$。我们假设$u_i(i=1, \ldots, I)$在$t_i$中严格递增，$u_0$在每个$t_i$中递减，并且这些函数是两次连续可微的。

给定(取决于类型的)分配${y(\theta)}{\theta \in \Theta}$，类型为$\theta_i$的代理$i(i=1, \ldots, I)$具有预期的或“临时”效用 $$ U_i\left(\theta_i\right) \equiv \mathrm{F}\theta{ }_i\left[u_i\left(y\left(\theta_i, \theta{-i}\right), \theta_i, \theta{-i}\right) \mid \theta_i\right]
$$
委托人有预期效用
$$
\mathrm{I}_{i 0} u_0(y(\theta), \theta) .
$$

经济代写|博弈论代写Game Theory代考|Mechanism Design with a Single Agent

下面的方法首先由Mirrlees(1971)提出，并被Mussa和Rosen(1978)、Baron和Myerson(1982)、Maskin和Riley (1984a)等人扩展和应用到不同的语境中。介绍，包括命题，遵循一般分析的Guesnerie和Laffont(1984)。 ${ }^{11}$

因为只有一个代理，所以我们省略了传输$(t)$上的下标，并在本节中键入$(\theta)$。我们假设代理的类型位于一个间隔$[\theta, \theta]$。代理知道$\theta$，而主体具有先验累积分布函数$P(P(\theta)=0, P(\theta)=1)$，具有可微密度$p(\theta)$，使得$p(\theta)>0$对于$[\theta, \bar{\theta}]$中的所有$\theta$。(密度的可微性不是必需的，但为方便起见，假设。)类型空间是单维的，${ }^{12}$但决策空间可能是多维的。(尽管我们考虑了多维决策的完整性，但读者可以从单维决策的情况中掌握主要思想。)(类型相关的)分配是一个从代理的类型到分配的函数:
$$
\theta \rightarrow y(\theta)=(x(\theta), t(\theta)) .
$$

经济代写|博弈论代写Game Theory代考请认准statistics-lab™

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

术语广义线性模型（GLM）通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归，以及方差分析和方差分析（仅含固定效应）。

有限元方法代写

有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。

有限元是一种通用的数值方法，用于解决两个或三个空间变量的偏微分方程（即一些边界值问题）。为了解决一个问题，有限元将一个大系统细分为更小、更简单的部分，称为有限元。这是通过在空间维度上的特定空间离散化来实现的，它是通过构建对象的网格来实现的：用于求解的数值域，它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统，以模拟整个问题。然后，有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。

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。

随机分析代写

随机微积分是数学的一个分支，对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。

时间序列分析代写

随机过程，是依赖于参数的一组随机变量的全体，参数通常是时间。随机变量是随机现象的数量表现，其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值（如1秒，5分钟，12小时，7天，1年），因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中，往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录，以得到其自身发展的规律。

回归分析代写

多元回归分析渐进（Multiple Regression Analysis Asymptotics）属于计量经济学领域，主要是一种数学上的统计分析方法，可以分析复杂情况下各影响因素的数学关系，在自然科学、社会和经济学等多个领域内应用广泛。

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，工具箱允许您学习和应用专业技术。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。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

经济代写|博弈论代写Game Theory代考|CS4

Posted on 2023年8月10日2023年8月28日 by statistics-lab

经济代写|博弈论代写Game Theory代考|Further Examples of Bayesian Equilibria

This section sketches the analyses of several Bayesian games. Although the first example is straightforward, the details of the other examples become somewhat involved, and many readers may wish to skip them. However, we refer to several of them in section 6.7.
Example 6.2: Cournot Competition with Incomplete Information Consider a duopoly playing Cournot (quantity) competition. Let firm $i$ ‘s profit be quadratic: $u_i=q_i\left(\theta_i-q_i-q_j\right)$, where $\theta_i$ is the difference between the intercept of the linear demand curve and firm i’s constant unit cost $(i-1,2)$ and where $q_i$ is the quantity chosen by firm $i\left(s_i=q_i\right)$. It is common knowledge that, for firm 1, $\theta_1=1$ (“firm 2 has complete information about firm 1,” or “firm 1 has only onc potential type”). Firm 2, however, has private information about its unit cost. Firm 1 believes that $\theta_2=\frac{3}{4}$ with probability $\frac{1}{2}$ and $\theta_2=\frac{5}{4}$ with probability $\frac{1}{2}$, and this belief is common knowledge. Thus, firm 2 has two potential types, which we will call the “low-cost type” $\left(\theta_2={ }_4^5\right)$ and the “high-cost type” $\left(\theta_2={ }_4^3\right)$. The two firms choose their outputs simultaneously.
l.ct us look for a pure-strategy equilibrium of this game. We denote firm I’s output by $q_1$, firm 2 ‘s output when $\theta_2=\frac{5}{4}$ by $q_2^{\mathrm{L}}$, and firm 2 ‘s output when $\theta_2-{ }_4^3$ by $q_2^{\mathrm{H}}$. Firm 2’s equilibrium choice $q_2\left(\theta_2\right)$ must satisfy
$$
q_2\left(\theta_2\right) \in \underset{4}{\arg \max }\left{q_2\left(\theta_2-q_1-q_2\right)\right} \Rightarrow q_2\left(\theta_2\right)=\left(\theta_2-q_1\right) / 2 .
$$
Firm I does not know which type of firm 2 it faces, so its payoff is the expected value over firm 2’s lypes:
$$
\begin{aligned}
& q_1 \subset \underset{\psi_1}{\arg \max }\left{\frac{1}{2} q_1\left(1-q_1-q_2^{\mathrm{H}}\right)+\frac{1}{2} q_1\left(1-q_1-q_2^{\mathrm{L}}\right)\right} \
& \Rightarrow q_1=-\begin{array}{r}
2-q_2^{\mathrm{H}}-q_2^{\mathrm{L}} \
4
\end{array} .
\end{aligned}
$$

经济代写|博弈论代写Game Theory代考|Interim vs. Ex Ante Dominance

If player $i$, instead of knowing the type-contingent strategies of his opponents, must try to predict them, then player $i$ must be concerned with how player $j \neq i$ thinks player $i$ would play for each possible type player $i$ might have. And player $i$ must also try to estimate player $j$ ‘s beliefs about player $i$ ‘s type, in order to predict the distribution of strategies that player $i$ expects to face.

This brings us to the question of how the players predict their opponents’ strategies, which in turn raises the following question: Should different types $\theta_1$ and $\theta_1^{\prime}$ of player 1 be viewed simply as a way of describing different information sets of a single player 1 , who makes a type-contingent decision at the ex ante stage (that is, before he learns his type)? This interpretation seems natural in the Harsanyi formulation, which introduces a move by nature that determines the “type” of a single player 1. Alternatively, should we think of $\theta_1$ and $\theta_1^{\prime}$ as denoting two different “individuals,” one of whom is selected by nature to “appear” when the game is played? In the first interpretation, the single $e x$ ante player 1 should be thought of as predicting his opponents’ play at the ex ante stage, so all types of player 1 would make the same prediction about the play of the other players. Under the second interpretation, the “different individuals” corresponding to different $\theta_1$ ‘s would each make their predictions at the “interim” stage (i.e., after learning their type), and the different types could make different predictions. (This second interpretation may become more plausible if we imagine that the “types” correspond to aspects of preferences that are genetically determined, for here the “ex ante” stage is difficult to interpret literally.)

It is inleresting to see that iterated strict dominance is at least as strong in the ex ante interpretation as in the interim interpretation and that the ex ante interpretation yields strictly stronger predictions in some games. To illustrate this, let us return to the public-good game of example 6.1. Using the interim approach to dominance, we ask which strategies are strictly dominated for player $i$ when his cost is $c_i$. Not contributing is not dominated for any positive cost level, as it is always better not to contribute if you expect that the opponent will contribute. However, if $c_i$ is greater than the private benefit of the good, which is 1 , then contributing is strictly dominated for player $i$.

博弈论代考

经济代写|博弈论代写Game Theory代考|Further Examples of Bayesian Equilibria

本节概述了几种贝叶斯博弈的分析。尽管第一个示例很简单，但其他示例的细节有些复杂，许多读者可能希望跳过它们。但是，我们将在第6.7节中提到其中的几个。
例6.2:不完全信息下的古诺竞争考虑一个双头垄断企业进行古诺(数量)竞争。坚定 $i$ 的利润是二次的: $u_i=q_i\left(\theta_i-q_i-q_j\right)$，其中 $\theta_i$ 直线需求曲线的截距和公司i不变单位成本的差是多少 $(i-1,2)$ 在哪里? $q_i$ 数量是由公司选择的吗 $i\left(s_i=q_i\right)$．众所周知，对于公司1来说， $\theta_1=1$ (“公司2有关于公司1的完整信息”或“公司1只有一种潜在类型”)。然而，公司2拥有关于其单位成本的私人信息。公司1认为 $\theta_2=\frac{3}{4}$ 有概率地 $\frac{1}{2}$ 和 $\theta_2=\frac{5}{4}$ 有概率地 $\frac{1}{2}$，这种信念是常识。因此，公司2有两种潜在类型，我们称之为“低成本型” $\left(\theta_2={ }_4^5\right)$ 以及“高成本型” $\left(\theta_2={ }_4^3\right)$．两家公司同时选择它们的产出。
让我们寻找这个博弈的纯策略均衡。我们用。表示公司I的输出 $q_1$时，公司2的产量 $\theta_2=\frac{5}{4}$ 通过 $q_2^{\mathrm{L}}$时，公司2的产量为 $\theta_2-{ }_4^3$ 通过 $q_2^{\mathrm{H}}$．公司2的均衡选择 $q_2\left(\theta_2\right)$ 必须满足
$$
q_2\left(\theta_2\right) \in \underset{4}{\arg \max }\left{q_2\left(\theta_2-q_1-q_2\right)\right} \Rightarrow q_2\left(\theta_2\right)=\left(\theta_2-q_1\right) / 2 .
$$
公司I不知道它面对的是哪种类型的公司2，所以它的收益是公司2类型的期望值:
$$
\begin{aligned}
& q_1 \subset \underset{\psi_1}{\arg \max }\left{\frac{1}{2} q_1\left(1-q_1-q_2^{\mathrm{H}}\right)+\frac{1}{2} q_1\left(1-q_1-q_2^{\mathrm{L}}\right)\right} \
& \Rightarrow q_1=-\begin{array}{r}
2-q_2^{\mathrm{H}}-q_2^{\mathrm{L}} \
4
\end{array} .
\end{aligned}
$$

经济代写|博弈论代写Game Theory代考|Interim vs. Ex Ante Dominance

如果玩家$i$不知道对手的类型随机策略，而是必须尝试预测它们，那么玩家$i$就必须关注玩家$j \neq i$认为玩家$i$会如何面对玩家$i$可能拥有的每种类型。参与人$i$还必须尝试估计参与人$j$对参与人$i$类型的信念，以便预测参与人$i$期望面对的策略分布。

这就引出了玩家如何预测对手策略的问题，这反过来又提出了以下问题:玩家1的不同类型$\theta_1$和$\theta_1^{\prime}$是否应该被简单地视为描述单个玩家1的不同信息集的一种方式，而玩家1在事前阶段(即在他了解自己的类型之前)做出了类型偶然决定?这种解释在Harsanyi公式中似乎很自然，它引入了一个决定单人玩家“类型”的自然移动。或者，我们是否应该认为$\theta_1$和$\theta_1^{\prime}$表示两个不同的“个体”，其中一个在玩游戏时自然选择“出现”?在第一种解释中，单个$e x$事前玩家1应该被认为是在事前阶段预测对手的玩法，所以所有类型的玩家1都会对其他玩家的玩法做出相同的预测。在第二种解释下，不同$\theta_1$对应的“不同个体”各自在“过渡”阶段(即在学习了自己的类型之后)做出预测，不同的类型可以做出不同的预测。(如果我们想象“类型”对应于基因决定的偏好方面，第二种解释可能会变得更合理，因为这里的“事前”阶段很难从字面上解释。)

有趣的是，迭代严格支配在事前解释中至少与在临时解释中一样强大，而且事前解释在某些游戏中产生了严格更强的预测。为了说明这一点，让我们回到示例6.1中的公共利益博弈。使用优势的临时方法，我们问当玩家$i$的成本为$c_i$时，哪种策略是严格劣势的。在任何正成本水平下，不做出贡献都不是主导，因为如果你期望对手做出贡献，那么不做出贡献总是更好的选择。然而，如果$c_i$大于物品的个人利益，即1，那么对于参与者$i$来说，贡献是严格支配的。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

经济代写|博弈论代写Game Theory代考|ECON314

Posted on 2023年8月10日2023年8月28日 by statistics-lab

经济代写|博弈论代写Game Theory代考|Changing the Information Structure with the I Ime Period

The folk theorem looks at a set of equilibrium payoffs as $\delta \rightarrow 1$, holding $\pi_y(a)$ constant. As we saw, whether the folk theorem holds depends on the amount of information the public outcome $y$ reveals. The interpretation of the result is therefore that almost all feasible, individually rational payoffs are equilibrium payoffs when $\delta$ is large in comparison with the information revealed by the outcome. Abreu, Pearce, and Milgrom (1990) show that the folk theorem need not hold if one interprets $\delta \rightarrow 1$ as the result of the interval between periods converging to 0 , and if the information revealed by $y$ deteriorates as the time interval shrinks. Why might this be the case? In games with observed actions, the public outcome is perfectly informative, and there is no reason to expect the information to change as the time period shrinks. In these games, then, we can interpret $\delta \rightarrow 1$ as a situation of either very little time preference or very short time periods. However, if players observe only imperfect signals of one another’s actions, it is plausible that the quality of their information depends on the length of each observation period. Thus, one cannot interpret the case of $\delta \cong 1$, with $\pi_v(a)$ fixed, as the study of what would occur if the time period became very short.

Abrcu, Pearce, and Milgrom (APM) investigate the effects of changing the time period and the associated information structure in two different examples. We will focus on a variant of their first example, a model of a repeated partnership game. We begin as usual by describing the stage game, which in the APM model is a continuous-time game of length $\tau$. The interpretation is that players lock in their actions at the start of the stage, and at the end of the stage the outcome and the payoffs are revealed. As in example 5.4, each player has two choices: work and shirk. Payoffs are chosen so that shirk is a dominant strategy in the stage game, and so that shirk is the minmax strategy. As in the example, the stage game has the structure of the prisoner’s dilemma: “Both shirk” is a Nash equilibrium in dominant strategies, and this equilibrium gives the players their minmax values. Payoffs arc normalized so that this minmax payoff is 0 , the (ex- pected) payoffs if both players work are $(c, c)$, and the payoff to shirking when the opponent works is $c+g$. (These are the expected payoffs, where the expectation is taken with respect to the corresponding distribution of output.) The difference between the APM stage game and example 5.4 is that, instead of there being only two outcomes each period (namely high and low output), the outcome is the number of “successes” in the period, which is distributed as a Poisson variable whose intensity is $\lambda$ if both players work and $\mu$ if one of them shirks, with $\lambda>\mu$. Thus, if the time period is short, it is unlikely that there will be more than one success, and the probability of one success in a period of length $d t$ is proportional to $d t$. This might correspond to a situation where the workers are trying to invent new products.

经济代写|博弈论代写Game Theory代考|Incomplete Information

When some players do not know the payoffs of the others, the game is said (o) have incomplete information. Many games of interest have incomplete information to at least some extent; the case of perfect knowledge of payoffs is a simplifying assumption that may be a good approximation in some cases.

As a particularly simple example of a game in which incomplete information matters, consider an industry with two firms: an incumbent (player 1) and a potential entrant (player 2). Player 1 decides whether to build a new plant, and simultaneously player 2 decides whether to enter. Imagine that player 2 is uncertain whether player 1 ‘s cost of building is 3 or 0 , while player 1 knows her own cost. The payoffs are depicted in figure 6.1. Player 2’s payoff depends on whether player 1 builds, but is not directly influenced by player l’s cost. Entering is profitable for player 2 if and only if player 1 does not build. Note also that player 1 has a dominant strategy: “build” if her cost is low and “don’t build” if her cost is high.

Let $p_1$ denote the prior probability player 2 assigns to player 1’s cost being high. Because player $I$ builds if and only if her cost is low, player 2 enters whenever $p_1>\frac{1}{2}$ and stays out if $p_1<\frac{1}{2}$. Thus, we can solve the game in figure 6.1 by the iterated deletion of strictly dominated strategies. Section 6.6 gives a careful analysis of iterated dominance arguments in games of incomplete information.

The analysis of the game becomes more complex when the low cost is only 1.5 instead of 0 , as in figure 6.2. In this new game, “don’t build” is still a dominant strategy for player 1 when her cost is high. However, when her cost is low, player l’s optimal strategy depends on her prediction of $y$, the probability that player 2 enters: Building is better than not building if
$$
1.5 y+3.5(1-y)>2 y+3(1-y),
$$
or
$$
y<\frac{1}{2} .
$$

博弈论代考

经济代写|博弈论代写Game Theory代考|Changing the Information Structure with the I Ime Period

民间定理将一组均衡收益视为$\delta \rightarrow 1$，保持$\pi_y(a)$不变。正如我们所看到的，民间定理是否成立取决于公共结果$y$所揭示的信息量。因此，对结果的解释是，当$\delta$与结果所揭示的信息相比较大时，几乎所有可行的、个体理性的收益都是均衡收益。Abreu, Pearce, and Milgrom(1990)表明，如果将$\delta \rightarrow 1$解释为周期间隔收敛于0的结果，并且$y$所揭示的信息随着时间间隔的缩小而恶化，则民间定理不必成立。为什么会这样呢?在观察行为的游戏中，公共结果是完全具有信息性的，没有理由期望信息会随着时间的缩短而改变。在这些游戏中，我们可以将$\delta \rightarrow 1$解释为时间偏好非常少或时间周期非常短的情况。然而，如果玩家只观察到对方行动的不完美信号，那么他们的信息质量就取决于每个观察期的长度。因此，我们不能把$\pi_v(a)$固定下来的$\delta \cong 1$案例解释为研究如果时间变得很短会发生什么。

Abrcu、Pearce和Milgrom (APM)在两个不同的例子中研究了改变时间周期和相关信息结构的影响。我们将关注他们第一个例子的一个变体，一个重复合作博弈的模型。我们像往常一样从描述阶段博弈开始，它在APM模型中是一个长度为$\tau$的连续时间博弈。其解释是，玩家在阶段开始时锁定自己的行动，在阶段结束时显示结果和收益。在例5.4中，每个玩家有两个选择:工作和逃避。收益的选择使得逃避是阶段博弈中的优势策略，因此逃避是最小最大策略。在这个例子中，阶段博弈具有囚徒困境的结构:“双方都逃避”是优势策略中的纳什均衡，这个均衡给了参与者最大最小值。收益是标准化的，所以这个最小最大收益是0，如果两个玩家都工作，(预期的)收益是$(c, c)$，当对手工作时，逃避的收益是$c+g$。(这些是预期收益，其中期望是相对于相应的产出分布的。)APM阶段博弈与示例5.4的不同之处在于，不同于每个阶段只有两个结果(即高输出和低输出)，结果是该时期“成功”的数量，它以泊松变量的形式分布，如果两个玩家都工作，强度为$\lambda$，如果其中一个玩家逃避，强度为$\mu$, $\lambda>\mu$。因此，如果时间段很短，则不太可能有多个成功，并且在长度为$d t$的时间段内一个成功的概率与$d t$成正比。这可能对应于工人试图发明新产品的情况。

经济代写|博弈论代写Game Theory代考|Incomplete Information

当一些参与者不知道其他参与者的收益时，这个博弈被称为(0)具有不完全信息。许多有趣的游戏至少在某种程度上具有不完整的信息;完全了解收益的情况是一个简化的假设，在某些情况下可能是一个很好的近似值。

作为一个特别简单的游戏例子，在这个游戏中，信息不完全很重要，考虑一个有两家公司的行业:现有的(参与人1)和潜在的进入者(参与人2)。参与人1决定是否建立一个新工厂，同时参与人2决定是否进入。想象一下，参与人2不确定参与人1的建造成本是3还是0，而参与人1知道自己的成本。结果如图6.1所示。玩家2的收益取决于玩家1是否建造，但并不直接受到玩家1成本的影响。当且仅当玩家1不进行建造时，玩家2才能够从中获利。还要注意的是，玩家1有一个优势策略:如果成本低就“建造”，如果成本高就“不建造”。

设$p_1$表示参与人2分配给参与人1的代价高的先验概率。因为当且仅当玩家$I$的成本较低时，玩家2便会在$p_1>\frac{1}{2}$时进入，并在$p_1<\frac{1}{2}$时离开。因此，我们可以通过迭代删除严格劣势策略来求解图6.1中的博弈。第6.6节详细分析了不完全信息博弈中的迭代优势论证。

当低成本仅为1.5而不是图6.2所示的0时，游戏分析将变得更加复杂。在这款新游戏中，当玩家1的成本很高时，“不建造”仍然是玩家1的主要策略。然而，当她的成本较低时，玩家1的最佳策略取决于她的预测$y$，即玩家2进入的概率:建造比不建造要好
$$
1.5 y+3.5(1-y)>2 y+3(1-y),
$$
或
$$
y<\frac{1}{2} .
$$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

经济代写|博弈论代写Game Theory代考|PSCI288

Posted on 2023年8月10日2023年8月28日 by statistics-lab

经济代写|博弈论代写Game Theory代考|Repeated Games with Imperfect Public Information

In the repeated games considered in the last section, each player observed the actions of the others at the end of each period. In many situations of economic interest this assumption is not satisfied, because the information that players receive is only an imperfect signal of the stage-game strategies of their opponents. Although there are many ways in which the assumption of observable actions can be relaxed, economists have focused on games of public information: At the end of each period, all players observe a “public outcome,” which is correlated with the vector of stage-game actions, and each player’s realized payoff depends only on his own action and the public outcome. Thus, the actions of a player’s opponents influence his payoff only through their influence on the distribution of outcomes. Games with observable actions are the special case where the public outcome consists of the realized actions themselves.

There are many examples of games in which the public outcome provides only imperfect information. Green and Porter (1984) published the first formal study of these games in the economics literature. Their model, which was intended to explain the occurrence of “price wars,” was motivated in part by the work of Stigler (1964). In Stigler’s model, cach firm observes its own sales but not the prices or quantities of its opponents. The aggregate level of consumer demand is stochastic. Thus, a fall in a firm’s sales might be due either to a fall in demand or to an unobserved price cut by an opponent. Since each firm’s only information about its opponents’ actions is its own level of realized sales, no firm knows what its opponents have observed, and there is no public information about the actions played. ${ }^{20}$ In contrast, the Grecn-Porter model does have public information, which makes it much easier to analyze. In that model, each firm’s payoff depends on its own output and on the publicly observed market price. Firms do not observe one another’s outputs, and the market price depends on an unobserved shock to demand as well as on aggregate output. Hence, an unexpectedly low market price could be due either to unexpectedly high output by an opponent or to unexpectedly low demand.

经济代写|博弈论代写Game Theory代考|The Model

In the repeated game, the public information at the beginning of period $t$ is
$$
h^{\prime}-\left(y^0, y^1, \ldots, y^2{ }^1\right) \text {. }
$$
Player $i$ also has private information at time $t$-namely, his own past choices of actions; denote this by $z_i^t$. A strategy for player $i$ is a scquence of maps from player $i$ ‘s time-t information to probability distributions over $A_1 ; \sigma_i^t\left(h^{\prime}, z_i^{\prime}\right)$ denotes the probability distribution chosen when player $i$ ‘s information is $\left(h^t, z_i^l\right)$.
Here are some illustrations of the model:

In a repeated game with observable actions, the set $Y$ of outcomes is isomorphic to the set $A$ of action profiles: $\pi_y(a)=1$ if $y$ is equivalent to $a$, and $\pi_v(a)=0$ otherwise.
In the Green-Porter model, $a_i \in[0, \bar{Q}]$ is firm $i$ ‘s output, and the outcome $y$ is the market price. Green and Porter make the additional assumptions that the probability distribution over outcomes depends only on the sum of the firms’ outputs and that every price has positive probability under every action profile.

博弈论代考

经济代写|博弈论代写Game Theory代考|Repeated Games with Imperfect Public Information

在上一节讨论的重复游戏中，每个玩家在每个阶段结束时观察其他人的行动。在许多经济利益的情况下，这个假设是不满足的，因为玩家收到的信息只是他们对手的阶段博弈策略的不完美信号。尽管有许多方法可以放松对可观察行为的假设，但经济学家关注的是公共信息博弈:在每个时期结束时，所有参与者都观察到一个“公共结果”，这与阶段博弈行为的向量相关，每个参与者的实现收益仅取决于他自己的行为和公共结果。因此，玩家对手的行为仅通过对结果分布的影响来影响玩家的收益。带有可观察行动的游戏是一种特殊情况，在这种情况下，公共结果由已实现的行动本身组成。

在许多游戏中，公共结果只能提供不完全信息。Green和Porter(1984)在经济学文献中首次发表了对这些博弈的正式研究。他们的模型旨在解释“价格战”的发生，其部分动机是Stigler(1964)的工作。在斯蒂格勒的模型中，每家公司都观察自己的销售情况，而不是竞争对手的价格或数量。消费者需求的总水平是随机的。因此，公司销售额的下降可能是由于需求的下降或竞争对手未察觉到的降价所致。由于每个公司对对手行动的唯一信息是自己的已实现销售水平，因此没有公司知道对手观察到了什么，也没有关于所采取行动的公开信息。${ }^{20}$相比之下，格林-波特模型确实有公共信息，这使得它更容易分析。在这个模型中，每个公司的收益取决于它自己的产出和公开观察到的市场价格。企业不观察彼此的产出，市场价格既取决于总产出，也取决于未被观察到的需求冲击。因此，意外的低市场价格可能是由于竞争对手意外的高产量或意外的低需求。

经济代写|博弈论代写Game Theory代考|The Model

在重复博弈中，周期开始时的公共信息$t$为
$$
h^{\prime}-\left(y^0, y^1, \ldots, y^2{ }^1\right) \text {. }
$$
玩家$i$在时间$t$上也有私人信息——也就是他自己过去的行为选择;用$z_i^t$表示。玩家$i$的策略是从玩家$i$的时间t信息到概率分布的一系列映射，$A_1 ; \sigma_i^t\left(h^{\prime}, z_i^{\prime}\right)$表示当玩家$i$的信息为$\left(h^t, z_i^l\right)$时所选择的概率分布。
以下是该模型的一些插图:

在具有可观察动作的重复博弈中，结果集$Y$与动作配置文件集$A$同构:如果$y$等同于$a$，则$\pi_y(a)=1$，否则等同于$\pi_v(a)=0$。

在Green-Porter模型中，$a_i \in[0, \bar{Q}]$是企业$i$的产出，结果$y$是市场价格。格林和波特还提出了另一个假设，即结果的概率分布只取决于企业产出的总和，而且在每种行为模式下，每种价格都有正概率。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

经济代写|博弈论代写Game Theory代考|ECN614

Posted on 2023年7月26日2023年8月25日 by statistics-lab

经济代写|博弈论代写Game Theory代考|Implementable Decisions and Allocations

A decision function $x: \theta \rightarrow \mathscr{X}$ is implementable if there exists a transfer function $t(\cdot)$ such that the allocation $y(\theta)=(x(\theta), t(\theta))$ for $\theta \in$ $[\theta, \theta]$ satisfies the incentive-compatibility constraint
(IC) $u_1(y(\theta), \theta) \geq u_1(y(\hat{\theta}), \theta)$ for all $(\theta, \hat{\theta}) \in[\theta, \bar{\theta}] \times[\theta, \bar{\theta}]$.
We will then say that the allocation $y(\cdot)$ is implementable.
Note that we ignore the individual-rationality constraint (that the agent be willing to participate in step 2) in this definition. Such a constraint, if any, must be reintroduced at the optimization stage.

Remark If $x(\cdot)$ is implementable through transfer $t(\cdot)$, there cxists an “indirect” or “fiscal” mechanism $t=T(x)$, in which the agent chooses a decision $x$, rather than an announcement of his type, that implements the same allocation. Consider the following scheme:

$$
T(x) \equiv \begin{cases}t & \text { if } \exists \hat{\theta} \text { such that } t=t(\hat{\theta}) \text { and } x=x(\hat{\theta}) \ \text { (if there exist several such } \hat{\theta} \text {, pick one) } \ -x & \text { otherwise. }\end{cases}
$$
Choosing an $x$ is de facto equivalent to announcing a $\hat{\theta}$.
We restrict our attention to decision profiles $x(\cdot)$ that are piecewise continuously differentiablc (“piecewise $C^1$ )). ${ }^{13}$ We now derive a necessary condition for $x(\cdot)$ to be implementable.

经济代写|博弈论代写Game Theory代考|Optimal Mechanisms

Now that we have characterized the set of implementable allocations, we can determine the optimal one for the principal. To do so, we must reintroduce the individual-rationality constraint for the agent. An implementable allocation that satisfics the individual-rationality constraint is called feasible; the principal’s problem is to choose the feasible allocation with the highest expected payoff. For simplicity, we assume that the agent’s reservation utility (i.e., his expected utility when he rejects the principal’s mechanism) is independent of his type.

A3 The reservation utility $u$ is independent of type; i.e., the participation constraint is
(IR) $u_1(x(\theta), t(\theta), \theta) \geq u$ for all $\theta$.
Under this assumption, if $u_1$ increases with the type ( $\left.\hat{\partial} u_1 / \partial \theta>0\right)$, then IR can bind only at $\theta=\theta$ : Any type $\theta>\theta$ can always announce $\hat{\theta}=\theta$, which gives him more than type $\theta$ ‘s utility, which is at least $u \cdot{ }^{18}$ For notational simplicity, we normalize $u=0$.

博弈论代考

经济代写|博弈论代写Game Theory代考|Implementable Decisions and Allocations

如果存在传递函数$t(\cdot)$，使得$\theta \in$$[\theta, \theta]$的分配$y(\theta)=(x(\theta), t(\theta))$满足激励兼容性约束，则决策函数$x: \theta \rightarrow \mathscr{X}$是可实现的
(IC) $u_1(y(\theta), \theta) \geq u_1(y(\hat{\theta}), \theta)$为所有$(\theta, \hat{\theta}) \in[\theta, \bar{\theta}] \times[\theta, \bar{\theta}]$。
然后我们说分配$y(\cdot)$是可实现的。
注意，在这个定义中，我们忽略了个体理性约束(即代理愿意参与步骤2)。这样的约束(如果有的话)必须在优化阶段重新引入。

注:如果$x(\cdot)$可以通过转移实现$t(\cdot)$，则存在一种“间接”或“财政”机制$t=T(x)$，在这种机制中，代理选择一个决策$x$，而不是他的类型的公告，实现相同的分配。考虑以下方案:

$$
T(x) \equiv \begin{cases}t & \text { if } \exists \hat{\theta} \text { such that } t=t(\hat{\theta}) \text { and } x=x(\hat{\theta}) \ \text { (if there exist several such } \hat{\theta} \text {, pick one) } \ -x & \text { otherwise. }\end{cases}
$$
选择一个$x$实际上相当于宣布一个$\hat{\theta}$。
我们将注意力限制在分段连续可微分的决策概要$x(\cdot)$(“分段$C^1$”)上。${ }^{13}$现在我们推导出$x(\cdot)$可实现的必要条件。

经济代写|博弈论代写Game Theory代考|Optimal Mechanisms

既然我们已经描述了一组可实现的分配，我们可以确定主体的最优分配。要做到这一点，我们必须重新引入个体理性约束。满足个体理性约束的可实现分配称为可行分配;委托人的问题是选择具有最高预期收益的可行分配。为简单起见，我们假设代理的保留效用(即，当他拒绝委托人的机制时，他的期望效用)与他的类型无关。

A3预约工具$u$与类型无关;即参与约束为
(IR) $u_1(x(\theta), t(\theta), \theta) \geq u$为所有$\theta$。
在这个假设下，如果$u_1$随着类型($\left.\hat{\partial} u_1 / \partial \theta>0\right)$)的增加而增加，那么IR只能绑定到$\theta=\theta$:任何类型$\theta>\theta$总是可以声明$\hat{\theta}=\theta$，这给了他比类型$\theta$的实用程序更多的东西，这至少是$u \cdot{ }^{18}$为了表示简单，我们规范化了$u=0$。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

经济代写|博弈论代写Game Theory代考|S159

Posted on 2023年7月26日2023年8月25日 by statistics-lab

经济代写|博弈论代写Game Theory代考|Incomplete Information

When some players do not know the payoffs of the others, the game is said to have incomplete information. Many games of interest have incomplete information to at least some extent; the case of perfect knowledge of payoffs is a simplifying assumption that may be a good approximation in some cases.

As a particularly simple example of a game in which incomplete information matters, consider an industry with two firms: an incumbent (player 1) and a potential entrant (player 2). Player 1 decides whether to build a new plant, and simultaneously player 2 decides whether to enter. Imagine that player 2 is uncertain whether player 1 ‘s cost of building is 3 or 0 , while player 1 knows her own cost. The payoffs are depicted in figure 6.1. Player 2 ‘s payoff depends on whether player 1 builds, but is not directly influenced by player 1 ‘s cost. Entering is profitable for player 2 if and only if player 1 does not build. Note also that player 1 has a dominant strategy: “build” if her cost is low and “don’t build” if her cost is high.

Let $p_1$ denote the prior probability player 2 assigns to player 1’s cost being high. Because player $\mathrm{I}$ builds if and only if her cost is low, player 2 enters whenever $p_1>\frac{1}{2}$ and stays out if $p_1<\frac{1}{2}$. Thus, we can solve the game in figure 6.1 by the iterated deletion of strictly dominated strategies. Section 6.6 gives a careful analysis of iterated dominance arguments in games of incomplete information.

经济代写|博弈论代写Game Theory代考|Providing a Public Good under Incomplete Information

The supply of a public good gives rise to the celebrated free-rider problem. Each player benefits when the public good is provided, but each would prefer the other players to incur the cost of supplying it. There are numerous variants of the public-good paradigm; we consider one studied experimentally by Palfrey and Rosenthal (1989). There are two players, $i=1,2$. Players decide simultaneously whether to contribute to the public good. and contributing is a $0-1$ decision. Each player derives a benefit of 1 if at least one of them provides the public good and 0 if none does; player $i$ ‘s cost of contributing is $c_i$. The payoffs are depicted in figure $6.4 .^2$

The benefits of the public good-1 each-are common knowledge, but each player’s cost is known only to that player. However, both players believe it is common knowledge that the $c_i$ are drawn independently from the same continuous and strictly increasing cumulative distribution function, $P(\cdot)$, on $[c, \bar{c}]$, where $c<1<\bar{c}$ (so $P(c)=0$ and $P(\bar{c})=1$ ). The cost $r_i$ is player $i$ ‘s “type.”

A pure strategy in this game is a function $s_i\left(c_i\right)$ from $[c, \bar{c}]$ into ${0,1}$, where 1 means “contribute” and 0 means “don’t contribute.” Player $i$ ‘s payoff is
$$
u_i\left(s_i, s_j, c_i\right)=\max \left(s_1, s_2\right)-c_i s_i .
$$
(Note that player $i$ ‘s payoff does not depend $c_j, j \neq i$ )

博弈论代考

经济代写|博弈论代写Game Theory代考|Incomplete Information

当一些玩家不知道其他玩家的收益时，这个游戏就被称为不完全信息。许多有趣的游戏至少在某种程度上具有不完整的信息;完全了解收益的情况是一个简化的假设，在某些情况下可能是一个很好的近似值。

设$p_1$表示参与人2分配给参与人1的代价高的先验概率。因为当且仅当玩家$\mathrm{I}$的成本较低时，玩家2便会在$p_1>\frac{1}{2}$时进入，并在$p_1<\frac{1}{2}$时离开。因此，我们可以通过迭代删除严格劣势策略来求解图6.1中的博弈。第6.6节详细分析了不完全信息博弈中的迭代优势论证。

经济代写|博弈论代写Game Theory代考|Providing a Public Good under Incomplete Information

公共产品的供给导致了著名的搭便车问题。当提供公共物品时，每个参与者都受益，但每个参与者都希望其他参与者承担提供公共物品的成本。公共利益范式有许多变体;我们以Palfrey和Rosenthal(1989)的实验研究为例。有两个玩家，$i=1,2$。参与者同时决定是否为公共利益做出贡献。贡献是一个$0-1$的决定。如果至少有一方提供公共物品，每个参与者的收益为1，如果没有提供，则为0;玩家$i$的贡献成本为$c_i$。收益如图所示 $6.4 .^2$

公共产品的利益——每个人——是众所周知的，但每个参与者的成本只有该参与者知道。然而，双方都认为这是常识，$c_i$是独立于相同的连续和严格增加的累积分布函数$P(\cdot)$，在$[c, \bar{c}]$上，$c<1<\bar{c}$(所以$P(c)=0$和$P(\bar{c})=1$)。成本$r_i$是玩家$i$的“类型”。

在这个游戏中，纯策略是一种功能 $s_i\left(c_i\right)$ 从 $[c, \bar{c}]$ 进入 ${0,1}$， 1表示“贡献”，0表示“不贡献”。玩家 $i$ 的回报是
$$
u_i\left(s_i, s_j, c_i\right)=\max \left(s_1, s_2\right)-c_i s_i .
$$
(注意玩家 $i$ 的回报并不取决于 $c_j, j \neq i$ ）

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

经济代写|博弈论代写Game Theory代考|E-327

Posted on 2023年7月26日2023年8月25日 by statistics-lab

经济代写|博弈论代写Game Theory代考|The Model

In the stage game, each player $i=1, \ldots, I$ simultaneously chooses a strategy $a_i$ from a finite set $A_i$. Each action profile $a \in A=\times_i A_i$ induces a probability distribution over the publicly observed outcomes $y$, which lie in a finite set $Y$. Let $\pi_y(a)$ denote the probability of outcome $y$ under $a$, and let $\pi(a)$ denote the probability distribution, which we will sometimes view as a row vector. Player $i$ ‘s realized payoff, $r_i\left(a_i, y\right)$, is independent of the actions of other players. (Otherwise, player $i$ ‘s payoff could give him private information about his opponents’ play.) Player i’s expected payoff under strategy profile $a$ is
$$
g_{\mathrm{i}}(a)=\sum_{\mathrm{y}} \pi_y(a) r_{\mathrm{i}}\left(a_i, y\right) .
$$
The payoffs and distributions over outcomes corresponding to mixed strategies $x$ are defined in the obvious way.
In the repeated game, the public information at the beginning of period $t$ is
$$
h^{\prime}-\left(y^0, y^1, \ldots, y^t{ }^1\right) \text {. }
$$
Player $i$ also has private information at time $t$-namely, his own past choices of actions; denote this by $z_i^t$. A strategy for player $i$ is a sequence of maps from player $i$ ‘s time-t information to probability distributions over $A_1 ; \sigma_i^i\left(h^{\prime}, z_i^i\right)$ denotes the probability distribution chosen when player $i$ ‘s information is $\left(h^t, z_i^t\right)$.
Here are some illustrations of the model:

In a repcated game with observable actions, the set $Y$ of outcomes is isomorphic to the set $A$ of action profiles: $\pi_y(a)=1$ if $y$ is equivalent to $a$, and $\pi_y(a)=0$ otherwise.
In the Green-Porter model, $a_i \in[0, \bar{Q}]$ is firm $i$ ‘s output, and the outcome $y$ is the market price. Green and Porter make the additional assumptions that the probability distribution over outcomes depends only on the sum of the firms’ outputs and that every price has positive probability under every action profile.
In the repeated partnership model, $a_i$ is player $i$ ‘s effort level and $y$ is the realized output. In the model of Radner (1986) and Radner et al. (1986), $A_i$ is the set ${$ work, shirk $}$. Closely related is the repeated principalagent model of Radner $(1981,1985)$, where the principal’s action is an observed monetary transfer and the agent’s effort level is not observed. Here the outcome is the pair (output, transfer).

经济代写|博弈论代写Game Theory代考|Trigger-Price Strategies

In the analysis of their oligopoly model, Green and Porter (1984) focus on equilibria in “trigger-price strategies,” which generalize the trigger-strategy equilibria introduced by Friedman (1971). Suppose that the set of outcomes $Y$ are interpreted as prices, so thà $Y \subseteq \mathbb{R}$, and each firm’s output $a_i$ must lic in the interval $[0, \bar{Q}]$. Payoff functions are assumed to be symmetric and attention is restricted to equilibria where all players choose the same actions in every period – that is, $\sigma_i\left(h^t\right)=\sigma_j\left(h^r\right)$ for all $t$ and $h^t$. (Thus, the equilibria are “strongly symmetric” in the sense of subsection 5.1.3.) Trigger-price-stratcgy profiles are indexed by three parameters, $\hat{a}, \hat{y}$, and $\hat{T}$. In these profiles, play can be in one of two possible “phases.” In the “cooperative phase,” all firms produce the same output, $a$. Play remains in the cooperative phase as long as each period’s realized price $y^{\prime}$ is at least the “trigger price” $\hat{y}$. If $y^{\prime}<\hat{y}$, then play switches to a “punishment phase” for $\hat{T}$ periods. In this phase, the players play a static Nash equilibrium $a^*$ in each period, regardless of the realized outcomes; after the $\hat{T}$ periods end, play returns to the cooperative phase.

If we simply take $\hat{a}=a^$, the strategies prescribe that the static equilibrium $a^$ be played every period, which is clearly an equilibrium, so trigger-price equilibria exist. More gencrally, we can characterize the trigger-price equilibria as follows: For fixed $\hat{y}$ and $\hat{a}$, let
$$
\lambda(\hat{a})=\operatorname{Prob}\left(y^t \geq \hat{y} \mid \hat{a}\right)
$$
be the probability that the outcome is at least the trigger level when players use profile $\hat{a}$. For convenience, normalize the payoff of the static equilibrium $a^*$ to be 0 . Then the (normalized) payoff if players conform to the strategies is
$$
\hat{v}=(1-\delta) g(\hat{a})+\delta \lambda(\hat{a}) \hat{v}+\delta(1 \cdots \lambda(\hat{a})) \delta^{\hat{t}} \hat{x}
$$
so that
$$
\hat{r}=\begin{gathered}
(1-\delta) g(\hat{a}) \
1-\delta \lambda(\hat{a})-\delta^{\hat{T}+1}(1-\lambda(\hat{a}))
\end{gathered}
$$

博弈论代考

经济代写|博弈论代写Game Theory代考|The Model

在阶段博弈中，每个参与者$i=1, \ldots, I$同时从一个有限集合$A_i$中选择一个策略$a_i$。每个动作概要$a \in A=\times_i A_i$在公开观察到的结果$y$上推导出一个概率分布，它位于一个有限集合$Y$中。设$\pi_y(a)$表示$a$下结果$y$的概率，设$\pi(a)$表示概率分布，我们有时将其视为行向量。玩家$i$的实现收益$r_i\left(a_i, y\right)$独立于其他玩家的行为。(否则，玩家$i$的收益可能会让他获得关于对手玩法的私人信息。)参与人i在策略profile $a$下的预期收益是
$$
g_{\mathrm{i}}(a)=\sum_{\mathrm{y}} \pi_y(a) r_{\mathrm{i}}\left(a_i, y\right) .
$$
与混合策略$x$对应的结果的收益和分布以明显的方式定义。
在重复博弈中，周期开始时的公共信息$t$为
$$
h^{\prime}-\left(y^0, y^1, \ldots, y^t{ }^1\right) \text {. }
$$
玩家$i$在时间$t$上也有私人信息——也就是他自己过去的行为选择;用$z_i^t$表示。玩家$i$的策略是从玩家$i$的时间t信息到概率分布的一系列映射，$A_1 ; \sigma_i^i\left(h^{\prime}, z_i^i\right)$表示当玩家$i$的信息为$\left(h^t, z_i^t\right)$时选择的概率分布。
以下是该模型的一些插图:

在具有可观察动作的重复游戏中，结果集$Y$与动作配置文件集$A$是同构的:如果$y$等同于$a$，则为$\pi_y(a)=1$，否则为$\pi_y(a)=0$。

在重复伙伴模型中，$a_i$为参与人$i$的努力水平，$y$为实现的产出。在Radner(1986)和Radner et al.(1986)的模型中，$A_i$是集合${$功，shirk $}$。密切相关的是Radner $(1981,1985)$的重复委托代理模型，其中委托人的行为是观察到的货币转移，代理人的努力水平不被观察到。这里的结果是对(输出，转移)。

经济代写|博弈论代写Game Theory代考|Trigger-Price Strategies

在分析他们的寡头垄断模型时，Green和Porter(1984)将重点放在“触发价格策略”中的均衡上，这是对Friedman(1971)提出的触发策略均衡的推广。假设结果集$Y$被解释为价格，因此th$Y \subseteq \mathbb{R}$，每个公司的产出$a_i$必须在区间$[0, \bar{Q}]$内。假设收益函数是对称的，并且注意力被限制在均衡中，即所有参与者在每个时期都选择相同的行动——也就是说，对于所有$t$和$h^t$，都是$\sigma_i\left(h^t\right)=\sigma_j\left(h^r\right)$。(因此，在第5.1.3小节的意义上，均衡是“强对称的”。)触发价格策略配置文件是由三个参数，$\hat{a}, \hat{y}$和$\hat{T}$索引。在这些描述中，游戏可以处于两个可能的“阶段”之一。在“合作阶段”，所有企业的产出相同，$a$。只要每个周期的实现价格$y^{\prime}$至少是“触发价格”$\hat{y}$，游戏就会保持在合作阶段。如果是$y^{\prime}<\hat{y}$，那么游戏就会切换到$\hat{T}$阶段的“惩罚阶段”。在这一阶段，玩家在每个时期都玩一个静态纳什均衡$a^*$，而不管实现的结果如何;$\hat{T}$阶段结束后，游戏回到合作阶段。

如果我们简单地取$\hat{a}=a^$，策略规定静态均衡$a^$每个时期都要进行，这显然是一个均衡，所以触发价格均衡是存在的。更一般地说，我们可以这样描述触发价格均衡:对于固定的$\hat{y}$和$\hat{a}$，令
$$
\lambda(\hat{a})=\operatorname{Prob}\left(y^t \geq \hat{y} \mid \hat{a}\right)
$$
当玩家使用配置文件$\hat{a}$时，结果至少是触发级别的概率。为方便起见，将静态均衡$a^*$的收益归一化为0。如果玩家遵循策略，那么(标准化)收益是
$$
\hat{v}=(1-\delta) g(\hat{a})+\delta \lambda(\hat{a}) \hat{v}+\delta(1 \cdots \lambda(\hat{a})) \delta^{\hat{t}} \hat{x}
$$
如此……以至于……
$$
\hat{r}=\begin{gathered}
(1-\delta) g(\hat{a}) \
1-\delta \lambda(\hat{a})-\delta^{\hat{T}+1}(1-\lambda(\hat{a}))
\end{gathered}
$$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

经济代写|博弈论代写Game Theory代考|Finitely Repeated Games

Posted on 2023年7月10日2023年7月10日 by statistics-lab

经济代写|博弈论代写Game Theory代考|Finitely Repeated Games

These games represent the case of a fixed known horizon $T$. The strategy spaces at each $l=0,1, \ldots, T$ are as defined above; the utilities are usually taken to be the time average of the per-period payoffs. Allowing for a discount factor $\delta$ close to 1 will not change the conclusions we present.

The set of equilibria of a finitely repeated game can be very different from that of the corresponding infinitely repeated game, because the scheme of self-reinforcing rewards and punishments used in the folk theorem can unravel backward from the terminal date. The classic example of this is the repeated prisoner’s dilemma. As observed in chapter 4, with a fixed finite horizon “always defect” is the only subgame-perfect-equilibrium outcome. In fact, with a bit more work one can show this is the only Nash outcome:

Fix a Nash equilibrium $\sigma^$. Both players must cheat in the last period, $T$, for any history $h^T$ that has positive probability under $\sigma^$, since doing so increases their period- $T$ payoff and since there are no future periods in which they might be punished. Next, we claim that both players must defect in period $T-1$ for any history $h^{T-1}$ with positive probability: We have already established that both players will chcat in the last period along the equilibrium path, so in particular if player $i$ conforms to the equilibrium strategy in period $T-1$ his opponent will defect in the last period, and hence player $i$ has no incentive not to defect in period $T-1$. An induction argument completes the proof. This conclusion, though not pathological, relies on the fact that the static equilibrium gives the players exactly their minmax values, as the following theorem shows.

经济代写|博弈论代写Game Theory代考|Repeated Games with Long-Run and Short-Run Players

The first variant we will consider supposes that some of the players are long-run players, as in standard repeated games, while the roles corresponding to other “players” are filled by a sequence of short-run players, each of whom plays only once.
Example 5.1
Suppose that a single long-run firm faces a sequence of short-run consumers, each of whom plays only once but is informed of all previous play when choosing his actions. Each period, the consumer moves first, and chooses whether or not to purchase a good from the firm. If the consumer does not purchase, then both players receive a payoff of 0 . If the consumer decides to purchase, then the firm must decide whether to produce high or low quality. If it produces high quality, both players have a payoff of 1 ; if it produces low quality, the firm’s payoff is 2 and the consumer’s payoff is

This game is a simplified version of those considered by Dybvig and Spatt (1980), Klein and Lefler (1981), and Shapiro (1982). ${ }^{12}$ Simon (1951)

and Kreps (1986) use a similar game to analyze the employment relationship, and to argue that one reason for the existence of “firms” is precisely to provide a long-run player who can be induced to be trustworthy by the prospect of future rewards and punishments.

The following strategies are a subgame-perfect equilibrium of this game when the firm is sufficiently patient: The firm starts out producing high yuality cvery time a consumer purchases, and continues to do so as long as it has never produced low quality in the past. If ever the firm produces low quality, it produces low quality at every subsequent opportunity. The consumers start out purchasing the good from the firm, and continue to do so so long as the firm has never produced low quality. If ever the firm produces low quality, then no consumer ever purchases again. The consumer’s strategies are optimal because each consumer cares only about that period’s payoff, and thus should buy if and only if that period’s quality is expected to be high. The firm does incur a short-run cost by producing high quality. but when the firm is patient this cost is offset by the fear that producing low quality will drive away future consumers. Note that this equilibrium suggests why consumers may prefer to deal with a firm that is expected to remain in business for a while, as opposed to a “fly-by-night” firm for whom long-run considerations are unimportant.

博弈论代考

经济代写|博弈论代写Game Theory代考|Finitely Repeated Games

这些游戏代表了一个固定已知视界$T$的情况。每个$l=0,1, \ldots, T$上的策略空间如上所述;效用通常被认为是每期收益的时间平均值。允许接近1的贴现因子$\delta$不会改变我们提出的结论。

有限重复博弈的平衡集可能与相应的无限重复博弈的平衡集非常不同，因为民间定理中使用的自我强化奖惩方案可以从结束日期向后分解。典型的例子是重复囚徒困境。正如在第4章中所观察到的，在固定的有限视界下，“总是缺陷”是唯一的子博弈-完美均衡结果。事实上，只要多做一点工作，就可以证明这是唯一的纳什结果:

修复纳什均衡$\sigma^$。两个玩家都必须在最后一个时期$T$，对于任何历史$h^T$，在$\sigma^$下有正概率，因为这样做增加了他们的时期- $T$收益，因为没有未来的时期，他们可能会受到惩罚。接下来，我们声称对于任何历史$h^{T-1}$，两个参与者都必须在时期$T-1$以正概率叛变:我们已经确定两个参与者都将在最后一个时期沿着均衡路径进行投机，所以特别是如果参与者$i$在时期$T-1$符合均衡策略，他的对手将在最后一个时期叛变，因此参与者$i$没有动机不在$T-1$时期叛变。一个归纳论证完成了这个证明。这一结论虽然不是病态的，但却是基于静态平衡能够提供给玩家最大最小值这一事实，如下面的定理所示。

经济代写|博弈论代写Game Theory代考|Repeated Games with Long-Run and Short-Run Players

我们将考虑的第一种变体假设一些玩家是长期玩家，就像在标准的重复游戏中一样，而与其他“玩家”对应的角色则由一系列短期玩家填补，每个玩家只玩一次。
例5.1
假设一个长期企业面对一系列短期消费者，每个消费者只玩一次游戏，但在选择行动时被告知之前的所有游戏。每个时期，消费者首先行动，选择是否从公司购买商品。如果消费者不购买，那么双方的收益都是0。如果消费者决定购买，那么企业必须决定生产高质量还是低质量的产品。如果产出高质量的产品，双方的收益都是1;如果生产低质量产品，公司的收益是2，消费者的收益是

这个游戏是Dybvig和Spatt (1980)， Klein和Lefler(1981)以及Shapiro(1982)所考虑的游戏的简化版本。${ }^{12}$西蒙(1951)

和Kreps(1986)用一个类似的游戏来分析雇佣关系，并认为“公司”存在的一个原因正是提供一个长期的参与者，他们可以被未来的奖励和惩罚的前景诱导为值得信赖的。

当公司有足够的耐心时，以下策略是该博弈的子博弈完美均衡:每次消费者购买时，公司都会开始生产高质量的产品，并且只要它过去从未生产过低质量的产品，就会继续这样做。如果一家公司曾经生产过低质量的产品，那么它在随后的每一次机会中都会生产低质量的产品。消费者开始从公司购买商品，只要公司不生产低质量的产品，消费者就会继续购买。如果企业生产的产品质量很差，那么消费者就不会再购买。消费者的策略是最优的，因为每个消费者只关心那段时间的收益，因此当且仅当那段时间的质量预期很高时，他们才会购买。这家公司生产高质量的产品确实产生了短期成本。但是，如果企业有耐心，这种成本就会被担心生产低质量产品会赶走未来的消费者所抵消。值得注意的是，这种平衡表明了为什么消费者可能更愿意与那些有望维持一段时间的公司打交道，而不是那些对长期考虑不重要的“不可靠”的公司。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

经济代写|博弈论代写Game Theory代考|Preemption Games

Posted on 2023年7月10日2023年7月10日 by statistics-lab

经济代写|博弈论代写Game Theory代考|Preemption Games

Preemption games are a rough opposite to the war of attrition, with $L(\hat{t})>F(\hat{t})$ for some range of times $\hat{t}$. Here the specification of the payoff to simultaneous stopping, $B(\cdot)$, is more important than in the war of attrition, as if $L$ exceeds $F$ we might expect both players to stop simultaneously. One example of a preemption game is the decision of whether and when to build a new plant or adopt a new innovation when the market is big enough to support only one such addition (Reinganum 1981a,b; Fudenberg and Tirole 1985). In this case $B(t)$ is often less than $F(t)$, as it cian be better to let an opponent have a monopoly than to incur duopoly losses.
()ne very stylized preemption game is “grab the dollar.” In this stationary game, time is discrete $(t=0,1, \ldots)$ and there is a dollar on the tablc, which either or both of the players can try to grab. If only one player grabs, that player receives 1 and the other 0 ; if both try to grab at once, the dollar is destroyed and both pay a fine of 1 ; if neither player grabs, the dollar remains on the table. The players use the common discount factor $\delta$, so that $l(t)=\delta^{\prime}, F(t)=0$, and $B(t)=-\delta^t$ for all $t$. Like the war of attrition, this game has asymmetric equilibria, where one player “wins” with probability 1. and also a symmetric mixed-strategy equilibrium, where each player grabs the dollar with probability $p^=\frac{1}{2}$ in cach period. (It is easy to check that this yields a symmetric equilibrium; to see that it is the only one, note that each player must be indifferent between stopping-i.e., grabbing-at date $t$, which yields payoff $\delta^{\prime}\left(\left(1-p^(t)\right)-p^(t)\right)$ if the other has not stopped before date $t$ and 0 otherwise, and never stopping, which yields payoff 0 , so that $p^(t)$ must equal $\frac{1}{2}$ for all $t$.) The payoffs in the symmetric cquilibrium are $(0,0)$, and the distribution over outcomes is that the probability that player 1 alone stops first at $t$, the probability that player 2 alone stops first at $t$, and the probability that both players stop simultaneously at $l$ are all equal to $\left({ }_4^1\right)^{t+1}$. Note that these probabilities are independent of the per-period discount factor, $\delta$, and thus of the period length, $\Lambda$, in contrast to the war of attrition, where the probabilities were proportional to the period length. This makes finding a continuous-time representation of this game more difficult.

经济代写|博弈论代写Game Theory代考|Iterated Conditional Dominance and the Rubinstein Bargaining Game

The last two sections presented several examples of infinite-horizon games with unique equilibria. The uniqueness arguments there can be strengthened, in that these games have a unique profile that satisfies a weaker concept than subgame perfection.

Definition 4.2 In a multi-stage game with observed actions, action $a_i^t$ is conditionally dominated at stage $t$ given history $h^t$ if, in the subgame beginning at $h^{\prime}$, every strategy for player $i$ that assigns positive probability to $a_i^{\prime}$ is strictly dominated. Iterated conditional dominance is the process that, at each round, deletes every conditionally dominated action in every subgame, given the opponents’ strategies that have survived the previous rounds.

It is easy to check that itcrated conditional dominance coincides with subgame perfection in finite games of perfect information. In these games it also coincides with Pearce’s (1984) extensive-form rationalizability. In general multi-stage games, any action ruled out by iterated conditional dominance is also ruled out by extensive-form rationalizability, but the cxact relationship between the two concepts has not been determined.
In a game of imperfect information, iterated conditional dominance can be weaker than subgame perfection, as it does not assume that players forecast that an equilibrium will occur in every subgame. To illustrate this point, consider a one-stage, simultaneous-move game. Then iterated conditional dominance coincides with iterated strict dominance, subgame perfection coincides with Nash equilibrium, and iterated strict dominance is in general weaker than Nash equilibrium.

博弈论代考

经济代写|博弈论代写Game Theory代考|Preemption Games

抢占式游戏与消耗战截然相反 $L(\hat{t})>F(\hat{t})$ 在一段时间内 $\hat{t}$．这里是同时停止的收益说明， $B(\cdot)$，比在消耗战中更重要，仿佛 $L$ 超过 $F$ 我们可能会期望两个玩家同时停下来。抢占游戏的一个例子是，当市场足够大，只能支持一种新产品时，决定是否以及何时建立新工厂或采用新创新(Reinganum 1981a,b;Fudenberg and Tirole 1985)。在这种情况下 $B(t)$ 通常小于 $F(t)$因为让对手垄断总比造成双寡头垄断的损失要好。
一个非常程式化的抢占游戏是“抢钱”。在这个静止博弈中，时间是离散的 $(t=0,1, \ldots)$ 桌上有一美元，任何一方或双方都可以试着去抢。如果只有一个玩家抓到了，那么这个玩家得到1，而另一个玩家得到0;如果双方都试图同时抢钱，美元就会被销毁，双方都要支付1美元的罚款;如果双方都没有抢到，那美元就留在桌上。玩家使用常见的折现系数 $\delta$，所以 $l(t)=\delta^{\prime}, F(t)=0$，和 $B(t)=-\delta^t$ 对所有人 $t$．与消耗战一样，这款游戏也具有非对称均衡，即一方玩家以1的概率“获胜”。还有一个对称的混合策略均衡，每个玩家都有概率地获得1美元 $p^=\frac{1}{2}$ 在每个时期。(很容易检验这是否产生对称平衡;要看到它是唯一的，注意每个玩家必须在停止之间无动于衷。，抓——在日期 $t$，产生收益 $\delta^{\prime}\left(\left(1-p^(t)\right)-p^(t)\right)$ 如果对方在约会前没有停止 $t$ 否则为0，永不停止，收益为0，所以 $p^(t)$ 必须相等 $\frac{1}{2}$ 对所有人 $t$)对称均衡的收益是 $(0,0)$结果的分布是参与人1首先停在 $t$，参与人2首先停在。的概率 $t$，以及两个玩家同时停在 $l$ 都等于 $\left({ }_4^1\right)^{t+1}$．请注意，这些概率与每周期的折现系数无关， $\delta$，因此周期长度， $\Lambda$而在消耗战中，概率与时间长短成正比。这使得寻找游戏的连续时间表现变得更加困难。

经济代写|博弈论代写Game Theory代考|Iterated Conditional Dominance and the Rubinstein Bargaining Game

最后两节介绍了具有独特均衡的无限视界博弈的几个例子。这里的独特性论点可以得到加强，因为这些游戏具有独特的特征，能够满足比子游戏完美性更弱的概念。

定义4.2在一个具有观察到的动作的多阶段博弈中，如果在从$h^{\prime}$开始的子博弈中，玩家$i$分配给$a_i^{\prime}$正概率的每个策略都是严格劣势的，那么在给定历史的$t$阶段，动作$a_i^t$是有条件劣势的$h^t$。迭代条件优势是指在每个回合中，根据对手在前几轮中幸存下来的策略，删除每个子游戏中的每个条件优势行动的过程。

在完全信息有限博弈中，迭代条件优势与子博弈完美性是一致的。在这些游戏中，它也与Pearce(1984)的广泛形式合理化相吻合。在一般的多阶段游戏中，任何被迭代条件支配所排除的行动也会被广泛形式的合理性所排除，但这两个概念之间的确切关系尚未确定。
在信息不完全的博弈中，迭代条件支配可能比子博弈的完美性更弱，因为它不假设玩家预测每个子博弈都会出现均衡。为了说明这一点，我们考虑一个单阶段同步移动的游戏。然后迭代条件优势与迭代严格优势重合，子博弈完美性与纳什均衡重合，迭代严格优势一般弱于纳什均衡。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

经济代写|博弈论代写Game Theory代考|Iterated Strict Dominance and Nash Equilibrium

Posted on 2023年7月10日2023年7月10日 by statistics-lab

经济代写|博弈论代写Game Theory代考|Iterated Strict Dominance and Nash Equilibrium

If the extensive form is finite, so is the corresponding strategic form, and the Nash existence theorem yields the existence of a mixed-strategy equilibrium. The notion of iterated strict dominance extends to extensiveform games as well; however, as we mentioned above, this concept turns out to have little force in most extensive forms. The point is that a player cannot strictly prefer one action over another at an information set that is not reached given his opponents’ play.

Consider figure 3.14. Here, player 2’s strategy $\mathrm{R}$ is not strictly dominated, as it is as good as $\mathrm{L}$ when player 1 plays $\mathrm{U}$. Morcover, this fact is not “pathological.” It obtains for all strategic forms whose payoffs are derived from an extensive form with the tree on the left-hand side of the figure. That is, for any assignment of payoffs to the terminal nodes of the tree, the payoffs to (U, L) and (U, R) must be the same, as both strategy profiles lead to the same terminal node. This shows that the set of strategic-form payoffs of a fixed game tree is of lower dimension than the set of all payoffs of the corresponding strategic form, so theorems based on generic strategic-form payoffs (see chapter 12) do not apply. In particular, there can be an even number of Nash equilibria for an open set of extensive-form payoffs. The game illustrated in figure 3.14 has two Nash equilibria, (U, R) and (D, L), and this number is not changed if the extensive-form payoffs are slightly perturbed. The one case where the odd-number theorem of chapter 12 applies is to a simultaneous-move game such as that of figure 3.4 ; in such a game, each terminal node corresponds to a unique strategy profile. Put differently: In simultaneous-move games, every strategy profile reaches every information set, and so no player’s strategy can involve a choice that is not implemented given his opponents’ play.

Recall that a game of perfect information has all its information sets as singletons, as in the games illustrated in figures 3.3 and 3.14 .

经济代写|博弈论代写Game Theory代考|Backward Induction and Subgame Perfection

As we have seen, the strategic form can be used to represent arbitrarily complex extensive-form games, with the strategies of the strategic form being complete contingent plans of action in the extensive form. Thus, the concept of Nash equilibrium can be applied to all games, not only to games where players choose their actions simultancously. However, many game theorists doubt that Nash equilibrium is the right solution concept for general games. In this section we will present a first look at “equilibrium refinements,” which are designed to separate the “reasonable” Nash equilibria from the “unreasonable” ones. In particular, we will discuss the ideas of backward induction and “subgame perfection.” Chapters 4, 5 and 13 apply these ideas to some classes of games of interest to economists.

Selten (1965) was the first to argue that in general extensive games some of the Nash equilibria are “more reasonable” than others. He began with the example illustrated here in figure 3.14. This is a finite game of perfect information, and the backward-induction solution (that is, the one obtained using Kuhn’s algorithm) is that player 2 should play $\mathrm{L}$ if his information set is reached, and so player 1 should play D. Inspection of the strategic form corresponding to this game shows that there is another Nash equilibrium, where player 1 plays $\mathrm{U}$ and player 2 plays $\mathrm{R}$. The profile $(\mathrm{U}, \mathrm{R})$ is a Nash equilibrium because, given that player 1 plays U, player 2’s information set is not reached, and player 2 loses nothing by playing R. But Selten argued, and we agree, that this equilibrium is suspect. After all, if player 2 ‘s information set is reached, then, as long as player 2 is convinced that his payoffs are as specified in the figure, player 2 should play $\mathbf{L}$. And if we were player 2 , this is how we would play. Moreover, if we were player 1 , we would expect player 2 to play $\mathbf{L}$, and so we would play $\mathrm{D}$.

In the now-familiar language, the equilibrium (U,R) is not “credible,” because it relies on an “empty threat” by player 2 to play $R$. The threat is “empty” because player 2 would never wish to carry it out.

博弈论代考

经济代写|博弈论代写Game Theory代考|Iterated Strict Dominance and Nash Equilibrium

如果扩展形式是有限的，那么相应的策略形式也是有限的，纳什存在性定理给出了混合策略均衡的存在性。迭代严格支配的概念也延伸到广泛形式的游戏中;然而，正如我们上面提到的，这个概念在大多数广泛的形式中几乎没有力量。关键在于，玩家不能严格地选择一种行动，而不是另一种行动。

考虑图3.14。在这里，参与人2的策略$\mathrm{R}$不是严格劣势，因为当参与人1的策略$\mathrm{U}$时，它和$\mathrm{L}$一样好。此外，这个事实并不是“病态的”。它适用于所有战略形式，其收益由图左侧的树状图导出。也就是说，对于树的终端节点的任何收益分配，(U, L)和(U, R)的收益必须是相同的，因为两种策略profile都指向相同的终端节点。这表明固定博弈树的策略形式收益集合比相应策略形式的所有收益集合的维数更低，因此基于一般策略形式收益的定理(见第12章)不适用。特别地，对于一个开放的广泛形式收益集，可以有偶数个纳什均衡。图3.14所示的博弈有两个纳什均衡(U, R)和(D, L)，如果泛化形式的收益受到轻微干扰，这个数字不会改变。第12章的奇数定理适用于图3.4这样的同时移动博弈;在这样的博弈中，每个终端节点对应一个唯一的策略配置文件。换句话说:在同步移动游戏中，每个策略配置文件都涉及到每个信息集，所以没有玩家的策略可以包含不考虑对手玩法的选择。

回想一下，一个完全信息博弈的所有信息集都是单信息，如图3.3和3.14所示。

经济代写|博弈论代写Game Theory代考|Backward Induction and Subgame Perfection

正如我们所看到的，战略形式可以用来表示任意复杂的广泛形式博弈，战略形式的策略是广泛形式中完整的偶然行动计划。因此，纳什均衡的概念可以应用于所有游戏，而不仅仅是玩家同时选择行动的游戏。然而，许多博弈论家怀疑纳什均衡是否是一般博弈的正确解概念。在本节中，我们将首先介绍“均衡优化”，它旨在将“合理的”纳什均衡与“不合理的”纳什均衡区分开来。特别地，我们将讨论逆向归纳法和“子博弈完善”的思想。第4章、第5章和第13章将这些思想应用于经济学家感兴趣的一些游戏类。

Selten(1965)是第一个提出在一般广泛博弈中，某些纳什均衡比其他均衡“更合理”的人。他从图3.14所示的例子开始。这是一个完全信息的有限博弈，逆向归纳解(即使用库恩算法得到的解)是，如果达到参与人2的信息集，参与人2应该选择$\ mathm {L}$，因此参与人1应该选择d。检查该博弈对应的策略形式表明，存在另一个纳什均衡，其中参与人1选择$\ mathm {U}$，参与人2选择$\ mathm {R}$。配置文件$(\ mathm {U}， \ mathm {R})$是纳什均衡，因为假设参与人1选择U，参与人2的信息集没有达到，参与人2选择R不会损失任何东西。但塞尔滕认为，我们同意，这个均衡是可疑的。毕竟，如果达到参与人2的信息集，那么，只要参与人2确信他的收益如图中所示，参与人2就应该选择$\mathbf{L}$。如果我们是参与人2，我们会这样玩。此外，如果我们是参与人1，我们会期望参与人2玩$\mathbf{L}$，所以我们会玩$\mathbf{D}$。

用现在熟悉的语言来说，均衡(U,R)不是“可信的”，因为它依赖于玩家2的“空威胁”来玩R。威胁是“空的”，因为玩家2永远不会想要执行它。

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