分类：博弈论代写

澳洲代写｜ECC2610｜Game theory and strategic thinking博弈论和战略思维蒙纳士大学

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

statistics-lab^TM为您提供蒙纳士大学（Monash University）Game theory and strategic thinking博弈论和战略思维澳洲代写代考和辅导服务！

课程介绍：

Game theory offers a tool for strategic thinking. It can be thought of as the art of beating your rivals, knowing that they are trying to do the same to you. Individuals, firms, governments and nations behave strategically, for good and bad. Over the last few decades, game theory has been developed for the purpose of understanding social phenomena. It has become the major tool used by social scientists to understand, predict and regulate strategic interaction among agents who have conflicting interests. This unit provides an introduction to game theory with an emphasis on real-world cases, including applications in economics and business.

澳洲代写｜MTH3330 ｜Optimisation and operations research优化和运筹学蒙纳士大学

Game theory and strategic thinking博弈论和战略思维案例

问题 1.

3) Splitting the Dollar(s). Players 1 and 2 are bargaining over how to split $\$ 10$. Each player $i$ names an amount, $s_i$ between 0 and 10 for herself. These numbers do not have to be in whole dollar units. The choices are made simultaneously. Each player’s payoff is equal to her own money payoff. We will consider this game under two different rules. In both cases, if $s_1+s_2 \leq 10$ then the players get the amounts that they named (and the remainder, if any, is destroyed).
a) In the first case, if $s_1+s_2>10$ then both players get zero and the money is destroyed. What are the (pure strategy) Nash Equilibria of this game?

(a) Nash Equilibrium of this game is any combination of two numbers that sums up to 10 . Any combination that in sum exceeds 10 destroys value for both players. Any combination that sums up to a number less than 10 induces each player to regret not having asked for more.

问题 2.

b) In the second case, if $s_1+s_2>10$ and the amounts named are different, then the person who names the smaller amount gets that amount and the other person gets the remaining money. If $s_1+s_2>10$ and $s_1=s_2$ then both players get $\$ 5$. What are the (pure strategy) Nash Equilibria of this game?

(b) There is a unique Nash Equilibrium of the game, each player chooses 5. The logic above suggests $\left(s_1, s_2\right)$ cannot be an equilibrium if $s_1+s_2<10$. If $s_1+s_2 \geq 10$, then the player with the smaller amount can always get more by picking a number closer to the higher amount. For example, let’s imagine that players pick 7 and 8 , securing payoffs of 7 and 3 , respectively. In this situation, player 1 regrets not choosing 7.999. Finally, if both players choose the same number $(>5)$, each player will regret not picking slightly less. For example, let’s imagine that both players pick 7 , securing payoffs of 5 each. In this situation, each player would regret not picking 6.999, which yields a higher payoff than 5. Answers that rounded strategies to the closest cent were also fine.

问题 3.

Now suppose these two games are played with the extra rule that the named amounts have to be in whole dollar units. Does this change the (pure strategy) Nash Equilibria in either case?

(c) If amounts must be in whole dollars, then there are four equilibria: $(5,5),(6,5)$, $(5,6)$ and $(6,6)$. In all four cases, players get 5 each and cannot improve their payoffs further.

澳洲代写｜ECC2610｜Game theory and strategic thinking博弈论和战略思维蒙纳士大学

Attribute	Detail
Course Code	ECC2610
Course Title	Game theory and strategic thinking
Coordinating Unit	Introductory microeconomics
Semester	Second semester
Mode	On-campus
Delivery Location	Clayton
Number of Units	Not provided in the text
Pre-Requisites	ECB1101, ECC1000, ECF1100, ECS1101, ECW1101
Lecturers	Associate Professor Paola Labrecciosa

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|ECON4750

Posted on 2023年8月29日2023年8月29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|Backward Induction and Subgame Perfection

Io verify that a strategy profile of a multi-stage game with observed actions is subgame perfect, it suffices to check whether there are any historics $h^t$ where some player $i$ can gain by deviating from the actions prescribed by $s_i$ at $h^t$ and conforming to $s_i$ thereafter. Since this “one-stagc-deviation principle” is essentially the principle of optimality of dynamic programming, which is based on backward induction, it helps illustrate how sub game perfection extends the idea of backward induction. We split the observation into two parts, corresponding to finite- and infinite-horizon games; some readers may prefer to read the first proof and take the second one on faith, although both are quite simple. For notational simplicity, we state the principle for pure strategies; the mixcd-strategy counterpart is straightforward.

Theorem 4.1 (one-stage-deviation principle for finite-horizon games) In a finite multi-stage game with observed actions, strategy profile $s$ is subgame perfect if and only if it satisfies the one-stage-deviation condition that no player $i$ can gain by deviating from $s$ in a single stage and conforming to $s$ thereafter. More precisely, profile $s$ is subgame perfect if and only if there is no player $i$ and no strategy $\hat{s}i$ that agrees with $s_i$ except at a single $t$ and $h^t$, and such that $\hat{s}_i$ is a better response to $s{-i}$ than $s_i$ conditional on history $h^t$ being reached. ${ }^1$

Proof The necessity of the one-stage-deviation condition (“only if”) follows from the definition of subgame perfection. (Note that the one-stagedeviation condition is not necessary for Nash equilibrium, as a Nashequilibrium profile may prescribe suboptimal responses at histories that do not occur when the profile is played.) To see that the one-stage-deviation condition is sufficient, suppose to the contrary that profile $s$ satisfies the condition but is not subgame perfect. Then there is a stage $t$ and a history $h^t$ such that some player $i$ has a strategy $\hat{s}i$ that is a better response to $s{-i}$ than $s_i$ is in the subgame starting at $h^t$. Let $t$ be the largest $t^{\prime}$ such that, for some $h^{\prime}, \hat{s}_i\left(h^{\prime}\right) \neq s_i\left(h^{\prime}\right)$. The one-stage-deviation condition implies $\hat{t}>t$, and since the game is finite, $\hat{t}$ is finite as well. Now consider an alternative strategy $\tilde{s}_i$ that agrees with $\hat{s}_i$ at all $t<\hat{t}$ and follows $s_i$ from stage $\hat{t}$ on. Since $\hat{s}_i$ agrees with $s_i$ from $\hat{t}+1$ on, the one-stage-deviation condition implies that $\hat{s}_i$ is as good a response as $\hat{s}_i$ in every subgame starting at $\hat{t}$, so $\tilde{s}_i$ is as good a response as $\hat{S}_i$ in the subgame starting at $t$ with history $h^t$. If $\hat{t}=t+1$, then $\tilde{s}_i=s_i$, which contradicts the hypothesis that $\hat{s}_i$ improves on $s_i$. If $\hat{t}>t+1$, we construct a strategy that agrees with $\hat{s}_i$ until $\hat{t}-2$. and argue that it is as good a response as $\hat{s}_i$, and so on: The alleged sequence of improving deviations unravels from its endpoint.

经济代写|博弈论代写Game Theory代考|The Repeated Prisoner’s Dilemma

This section discusses the way in which repeated play introduces new equilibria by allowing players to condition their actions on the way their opponents played in previous periods. We begin with what is probably the best-known example of a repeated game: the celebrated “prisoner’s dilemma,” whose static version we discussed in chapter 1 . Suppose that the per-period payoffs depend only on current actions $\left(g_i\left(a^t\right)\right)$ and are as shown in figure 4.1 , and suppose that the players discount future payoffs with a common discount factor $\delta$. We will wish to consider how the equilibrium payoffs vary with the horizon $T$. To make the payoffs for different horizons comparable, we normalize to express them all in the units used for the per-period payoffs, so that the utility of a sequence $\left{a^0, \ldots, a^T\right}$ is
$$
\begin{gathered}
1-\delta \
1 \
\delta^{T+1}
\end{gathered} \sum_{i=0}^T \delta^t y_i\left(a^t\right) .
$$
This is called the “average discounted payoff.” Since the normalization is simply a rescaling, the normalized and present-value formulations represent the same preferences. The normalized versions makc it easier to see what happens as the discount factor and the time horizon vary, by measuring all payoffs in terms of per-period averages. For example, the present value of a flow of 1 per period from date 0 to date $T$ is $\left(1-\delta^{T+1}\right) /(1-\delta)$; the average discounted value of this flow is simply 1 .

We begin with the case in which the game is played only once. Then cooperating is strongly dominated, and the unique equilibrium is for both players to defect. If the game is repeated a finite number of times, subgame perfection requires both players to defect in the last period, and backward induction implies that the unique subgame-perfect equilibrium is for both players to defect in every period. $^2$

博弈论代考

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

为了验证带有观察到的行动的多阶段游戏的策略配置文件是子游戏完美的，它足以检查是否存在任何历史$h^t$，其中一些玩家$i$可以通过在$h^t$偏离$s_i$规定的行动并遵循$s_i$而获得收益。由于这种“一阶段偏差原则”本质上是基于逆向归纳法的动态规划的最优性原则，因此它有助于说明子博弈完美性如何扩展逆向归纳法的思想。我们将观察结果分成两部分，分别对应于有限视界和无限视界游戏;有些读者可能更喜欢读第一个证明，而相信第二个证明，尽管这两个证明都很简单。为了简化符号，我们陈述了纯策略的原则;混合策略的对应物很简单。

定理4.1(有限视界博弈的一阶段偏差原理)在一个有限的多阶段博弈中，当且仅当策略剖面$s$满足一阶段偏差条件，即任何参与者$i$都不能在某一阶段偏离$s$而随后服从$s$而获得收益。更准确地说，配置文件$s$是子博弈完美的，当且仅当没有玩家$i$，也没有策略$\hat{s}i$与$s_i$一致，除了$t$和$h^t$，并且$\hat{s}_i$是$s{-i}$比$s_i$更好的响应，条件是$h^t$已经到达。 ${ }^1$

单阶段偏差条件(“只有当”)的必要性从子博弈完美性的定义出发。(请注意，一级偏差条件对于纳什均衡来说是不必要的，因为纳什均衡剖面可能会规定在历史上的次优响应，而这些响应在剖面播放时不会发生。)为了证明一级偏差条件是充分的，假设曲线相反 $s$ 满足条件，但不是完美的子博弈。然后是一个舞台 $t$ 还有一段历史 $h^t$ 这样一些玩家 $i$ 有策略 $\hat{s}i$ 这是一个更好的回应 $s{-i}$ 比 $s_i$ 是在子游戏开始的时候 $h^t$．让 $t$ 做最大的 $t^{\prime}$ 这样，对一些人来说 $h^{\prime}, \hat{s}_i\left(h^{\prime}\right) \neq s_i\left(h^{\prime}\right)$．一级偏差条件表示 $\hat{t}>t$，由于这个博弈是有限的， $\hat{t}$ 也是有限的。现在考虑另一种策略 $\tilde{s}_i$ 这与 $\hat{s}_i$ 一点也不 $t<\hat{t}$ 然后是 $s_i$ 舞台上 $\hat{t}$ 继续。自从 $\hat{s}_i$ 同意 $s_i$ 从 $\hat{t}+1$ On，一级偏差条件意味着 $\hat{s}_i$ 是一个好的回应吗 $\hat{s}_i$ 在每一个子游戏中 $\hat{t}$所以 $\tilde{s}_i$ 是一个好的回应吗 $\hat{S}_i$ 在子游戏开始于 $t$ 有历史 $h^t$．如果 $\hat{t}=t+1$那么， $\tilde{s}_i=s_i$，这与假设相矛盾 $\hat{s}_i$ 改进于 $s_i$．如果 $\hat{t}>t+1$，我们就会制定一个与之一致的策略 $\hat{s}_i$ 直到 $\hat{t}-2$．并认为这是一个很好的回应 $\hat{s}_i$等等:所谓的改进偏差序列从它的端点开始解开。

经济代写|博弈论代写Game Theory代考|The Repeated Prisoner’s Dilemma

这一部分讨论了重复游戏如何通过允许玩家根据对手在前一阶段的行为来调整自己的行为，从而引入新的平衡。我们从重复博弈中最著名的例子开始:著名的“囚徒困境”，我们在第一章中讨论过它的静态版本。假设每个时期的收益只取决于当前的行为$\left(g_i\left(a^t\right)\right)$，如图4.1所示，并假设参与者用一个共同的贴现因子$\delta$来贴现未来的收益。我们希望考虑均衡收益如何随视界变化$T$。为了使不同层次的收益具有可比性，我们将它们归一化，用每个周期收益的单位来表示它们，因此序列$\left{a^0, \ldots, a^T\right}$的效用是
$$
\begin{gathered}
1-\delta \
1 \
\delta^{T+1}
\end{gathered} \sum_{i=0}^T \delta^t y_i\left(a^t\right) .
$$
这被称为“平均贴现收益”。由于归一化只是简单的重新缩放，因此归一化和现值公式表示相同的偏好。标准化的版本可以更容易地看到贴现因子和时间范围的变化，通过衡量每个时期的平均值的所有收益。例如，从日期0到日期$T$的每个期间1的流量的现值为$\left(1-\delta^{T+1}\right) /(1-\delta)$;这个流的平均折现值是1。

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

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
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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|BUS-G303

Posted on 2023年8月29日2023年8月29日 by statistics-lab

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 $\mathrm{Nash}$ 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
8 . The existence of an optimal choice from a compact set of actions requires that payoffs be upper semi-continuous in the choice made. (A real-valued function $f(x)$ is upper semicontinuous if $x^n \rightarrow x$ implies $\lim _{n \rightarrow x} f\left(x^n\right) \leq f(x)$.) 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 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 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 $\mathrm{L}$, and so we would play $\mathrm{D}$.

In the now-familiar language, the equilibrium $(\mathbf{U}, \mathbf{R})$ 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.

经济代写|博弈论代写Game Theory代考|C’ritiques of Backward Induction

Consider the 1 -player game illustrated in figure 3.18, where each player $i<I$ can either end the game by playing ” $D$ ” or play ” $A$ ” and give the move to player $i+1$. (To readers who skipped sections 3.3-3.5: Figure 3.18 depicts a “game tree.” Though you have not seen a formal definition of such trees, we trust that the particular trees we use in this subsection will be clear.) If player $i$ plays D, each player gets $1 / i$; if all players play A, cach gets 2 .

Since only onc player moves at a time, this is a game of perfect information, and we can apply the backward-induction algorithm, which predicts that all players should play $A$. If $I$ is small, this seems like a reasonable prediction. If $I$ is very large, then, as player 1 , we ourselves would play D and not A on the basis of a “robustness” argument similar to the one that suggested the inefficient equilibrium in the stag-hunt game of subsection 1.2 .4 .

First, the payoff 2 requires that all $I-1$ other players play $A$. If the probability that a given player plays $A$ is $p<1$, independent of the others. the probability that all $I-1$ other players play $\mathrm{A}$ is $p^I{ }^1$, which can be quite small even if $p$ is very large. Second, we would worry that player 2 might have these same concerns; that is, player 2 might play D to safeguard against either “mistakes” by future players or the possibility that player 3 might intentionally play $\mathrm{D}$.

A related observation is that longer chains of backward induction presume longer chains of the hypothesis that “player 1 knows that player 2 knows that player 3 knows… the payoffs.” If $I=2$ in figure 3.18 , backward induction supposes that player 1 knows player 2 ‘s payoff, or at least that player 1 is fairly sure that player 2 ‘s optimal choice is A. If $I=3$, not only must players 1 and 2 know player 3 ‘s payoff, in addition, player 1 must know that player 2 knows player 3’s payoff, so that player 1 can forecast player 2’s forecast of player 3’s play. If player 1 thinks that player 2 will forecast player 3 ‘s play incorrectly, then player 1 may choose to play D.

博弈论代考

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，$\mathrm{Nash}$均衡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解决问题的正确概念
8。从一组紧致的行动中存在最优选择，要求所作选择的收益是上半连续的。(如果$x^n \rightarrow x$暗示$\lim _{n \rightarrow x} f\left(x^n\right) \leq f(x)$，则实值函数$f(x)$是上半连续的。)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(即使用库恩算法获得的解)是，如果达到参与人2的信息集，参与人2应该选择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$\mathrm{U}$，参与人2选择$\mathrm{R}$。配置文件$(\mathrm{U}, \mathrm{R})$是纳什均衡，因为假设参与人1选择U，参与人2的信息集不会达到，参与人2选择$R$不会损失任何东西。但塞尔滕认为，这种均衡是可疑的，我们也同意这一点。毕竟，如果达到了参与人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l，如果我们是参与人2，我们就会这么玩。此外，如果我们是参与人1，我们会期望参与人2玩$\mathrm{L}$，所以我们会玩$\mathrm{D}$。

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

经济代写|博弈论代写Game Theory代考|C’ritiques of Backward Induction

考虑图3.18所示的1人博弈，其中每个玩家$i<I$可以通过“$D$”结束游戏，也可以通过“$A$”结束游戏，并将移动权交给玩家$i+1$。(对于跳过3.3-3.5节的读者:图3.18描绘了一个“游戏树”。虽然您还没有看到这种树的正式定义，但我们相信在本小节中使用的特定树将是清楚的。)如果玩家$i$选择D，每个玩家都得到$1 / i$;如果所有人都选A，每人得2分。

由于一次只有一个玩家移动，这是一个完全信息的博弈，我们可以应用反向归纳算法，该算法预测所有玩家都应该玩$A$。如果$I$很小，这似乎是一个合理的预测。如果$I$非常大，那么作为参与人1，我们自己就会选择D而不是A，这是基于“稳健性”论证，类似于第1.2 .4小节中提出的猎鹿博弈中的低效均衡。

首先，收益2要求所有$I-1$其他玩家都玩$A$。如果给定玩家选择$A$的概率是$p<1$，与其他玩家无关。所有$I-1$其他玩家都玩$\mathrm{A}$的概率是$p^I{ }^1$，即使$p$很大，这个概率也很小。其次，我们会担心玩家2也会有同样的担忧;也就是说，玩家2可能会选择D，以防止未来玩家的“错误”，或者玩家3可能会故意选择$\mathrm{D}$。

一个相关的观察是，更长的逆向归纳链假设了更长的假设链，即“参与人1知道参与人2知道参与人3知道……收益”。如果在图3.18中$I=2$，逆向归纳假设参与人1知道参与人2的收益，或者至少参与人1相当确定参与人2的最优选择是a。如果$I=3$，不仅参与人1和参与人2必须知道参与人3的收益，另外，参与人1必须知道参与人2知道参与人3的收益，这样参与人1就可以预测参与人2对参与人3的收益的预测。如果参与人1认为参与人2会错误地预测参与人3的策略，那么参与人1可能会选择D。

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|Econ411

Posted on 2023年8月29日2023年8月29日 by statistics-lab

经济代写|博弈论代写Game Theory代考|Behavior Strategies

This section defines strategies and equilibria in extensive-form games and relates them to strategies and equilibria of the strategic-form model. Let $H_i$ be the set of player $i$ ‘s information sets, and let $A_i \equiv \bigcup_{h_i \in H_i} A\left(h_i\right)$ be the set of all actions for player $i$. A pure strategy for player $i$ is a map $s_i: H_i \rightarrow A_i$, with $s_i\left(h_i\right) \in A\left(h_i\right)$ for all $h_i \in H_i$. Player $i$ ‘s space of pure strategies, $S_i$, is simply the space of all such $s_i$. Since each pure strategy is at map from information sets to actions, we can write $S_i$ as the Cartesian product of the action spaces at each $h_i$ :
$$
S_i=\underset{h_i \in H_i}{\times} A\left(h_i\right) .
$$
In the Stackelberg example of figure 3.3, player 1 has a single information set and three actions, so that he has three pure strategies. Player 2 has three information sets, corresponding to the three possible choices of player 1 , and player 2 has three possible actions at each information set, so player 2 has 27 pure strategies in all. More generally, the number of player $i$ ‘s pure strategies, # $S_i$, equals
$$
\prod_{h_i \in H_i} #\left(A\left(h_i\right)\right) \text {. }
$$
Given a pure strategy for each player $i$ and the probability distribution over Nature’s moves, we can compute a probability distribution over outcomes and thus assign expected payoffs $u_i(s)$ to each strategy profile $s$. The information sets that are reached with positive probability under profile $s$ are called the path of $s$.

Now that we have defined the payoffs to each pure strategy, we can proceed to define a pure-strategy Nash equilibrium for an extensive-form game as a strategy profile $s^$ such that each player $i$ ‘s strategy $s_i^$ maximizes his expected payoff given the strategies $s_{-i}^*$ of his opponents. Note that since the definition of Nash equilibrium holds the strategies of player $i$ ‘s opponents fixed in testing whether player $i$ wishes to deviate, it is as if the players choose their strategies simultaneously. This does not mean that in Nash equilibrium players necessarily choose their actions simultaneously. For example, if player 2’s fixed strategy in the Stackelberg game of figure 3.3 is the Cournot reaction function $\hat{s}_2=(4,4,3)$, then when player 1 treats player 2’s strategy as fixed he does not presume that player 2’s action is unaffected by his own, but rather that player 2 will respond to player 1’s action in the way specified by $\hat{S}_2$.

经济代写|博弈论代写Game Theory代考|The Strategic-Form Representation of Extensive-Form Games

Our next step is to relate extensive-form games and equilibria to the strategic-form model. To define a strategic form from an extensive form, we simply let the pure strategies $s \in S$ and the payoffs $u_i(s)$ be exactly those we defined in the extensive form. A different way of saying this is that the same pure strategies can be interpreted as either extensive-form or strategic-form objects. With the extensive-form interpretation, player $i$ “waits” until $h_i$ is reached before deciding how to play there; with the strategic-form interpretation, he makes a complete contingent plan in advance.

Figure 3.8 illustrates this passage from the extensive form to the strategic form in a simple example. We order player 2’s information sets from left to right, so that, for example, the strategy $s_2=(\mathrm{L}, \mathrm{R})$ means that he plays $L$ after $L^{\prime}$ and $R$ after D.

As another example, consider the Stackelberg game illustrated in figure 3.3. We will again order player 2’s information sets from left to right, so that player 2 ‘s strategy $\hat{s}_2=(4,4,3)$ means that he plays 4 in response to $q_1=3$, plays 4 in response to 4 , and plays 3 in response to 6 . (This strategy happens to be player 2’s Cournot reaction function.) Since player 2 has three information sets and three possible actions at each of these sets, he has 27 pure strategies. We trust that the reader will forgive our not displaying the strategic form in a matrix diagram!

There can be several extensive forms with the same strategic form, as the example of simultaneous moves shows: Figures $3.4 \mathrm{a}$ and $3.4 \mathrm{~b}$ both correspond to the same strategic form for the Cournot game.

At this point we should note that the strategy space as we have defined it may be unnecessarily large, as it may contain pairs of strategies that are “equivalent” in the sense of having the same consequences regardless of how the opponents play.

博弈论代考

经济代写|博弈论代写Game Theory代考|Behavior Strategies

本节定义了广泛形式博弈中的策略和均衡，并将它们与策略形式模型中的策略和均衡联系起来。设$H_i$为玩家$i$的信息集合，设$A_i \equiv \bigcup_{h_i \in H_i} A\left(h_i\right)$为玩家$i$的所有动作集合。玩家$i$的纯策略是一张地图$s_i: H_i \rightarrow A_i$, $s_i\left(h_i\right) \in A\left(h_i\right)$代表所有的$h_i \in H_i$。参与人$i$的纯策略空间$S_i$，就是所有这些的空间$s_i$。由于每个纯策略都是从信息集映射到行动，我们可以将$S_i$写成每个$h_i$处的行动空间的笛卡尔积:
$$
S_i=\underset{h_i \in H_i}{\times} A\left(h_i\right) .
$$
在图3.3的Stackelberg例子中，参与人1有一个信息集和三个行动，所以他有三个纯策略。参与人2有三个信息集，对应于参与人1的三种可能选择，参与人2在每个信息集有三种可能的行动，因此参与人2总共有27种纯策略。更一般地说，玩家$i$的纯策略数量，＃ $S_i$等于
$$
\prod_{h_i \in H_i} #\left(A\left(h_i\right)\right) \text {. }
$$
给定每个玩家的纯策略$i$和自然移动的概率分布，我们可以计算结果的概率分布，从而为每个策略配置文件$s$分配预期收益$u_i(s)$。在概要文件$s$下以正概率到达的信息集称为$s$的路径。

既然我们已经定义了每种纯策略的收益，我们就可以将广义博弈的纯策略纳什均衡定义为策略概要$s^$，使得每个参与人$i$的策略$s_i^$在给定其对手的策略$s_{-i}^*$的情况下最大化其预期收益。请注意，由于纳什均衡的定义在测试参与人$i$是否希望偏离时，将参与人$i$对手的策略固定下来，这就好像参与人同时选择了他们的策略。这并不意味着在纳什均衡中，参与者必须同时选择他们的行动。例如，如果在图3.3的Stackelberg博弈中，参与人2的固定策略是古诺反应函数$\hat{s}_2=(4,4,3)$，那么当参与人1将参与人2的策略视为固定策略时，他不会假设参与人2的行动不受自己的影响，而是认为参与人2将以$\hat{S}_2$指定的方式对参与人1的行动做出反应。

经济代写|博弈论代写Game Theory代考|The Strategic-Form Representation of Extensive-Form Games

我们的下一步是将广泛形式的博弈和均衡与战略形式的模型联系起来。为了从扩展形式中定义策略形式，我们简单地让纯策略$s \in S$和收益$u_i(s)$与我们在扩展形式中定义的完全相同。换句话说，同样的纯策略既可以被解释为外延形式客体，也可以被解释为策略形式客体。根据广义解释，玩家$i$“等待”到$h_i$才决定如何在那里玩;在战略形式的解释下，他提前制定了一个完整的应急计划。

图3.8用一个简单的例子说明了从外延形式到策略形式的转变。我们将参与人2的信息集从左到右排序，因此，例如，策略$s_2=(\mathrm{L}, \mathrm{R})$意味着他在$L^{\prime}$之后选择$L$，在D之后选择$R$。

另一个例子是图3.3所示的Stackelberg游戏。我们将再次对参与人2的信息集从左到右排序，因此参与人2的策略$\hat{s}_2=(4,4,3)$意味着他对$q_1=3$的回应是4，对4的回应是4，对6的回应是3。(这个策略恰好是参与人2的古诺反应函数。)因为参与人2有三个信息集，每个信息集有三个可能的行动，所以他有27个纯策略。我们相信读者会原谅我们没有在矩阵图中展示战略形式!

可以有几种具有相同策略形式的扩展形式，如同时移动的例子所示:图$3.4 \mathrm{a}$和$3.4 \mathrm{~b}$都对应于古诺博弈的相同策略形式。

在这一点上，我们应该注意到我们所定义的策略空间可能是不必要的大，因为它可能包含“等效”的策略对，即无论对手如何操作都具有相同的结果。

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14日2023年8月28日 by statistics-lab

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:
$$
\theta \rightarrow y(\theta)=(x(\theta), t(\theta)) .
$$

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|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 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
$$
g_{\mathrm{i}}(a)=\sum_y \pi_y(a) r_i\left(a_i, y\right) .
$$
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
$$
h^{\prime}-\left(y^0, y^1, \ldots, y^2{ }^1\right) \text {. }
$$
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
$$
g_{\mathrm{i}}(a)=\sum_y \pi_y(a) r_i\left(a_i, y\right) .
$$
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
$$
h^{\prime}-\left(y^0, y^1, \ldots, y^2{ }^1\right) \text {. }
$$
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。

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