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

statistics-labTM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辅导服务！

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.

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

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.

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.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

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

## 经济代写|博弈论代写Game Theory代考|ECON4750

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|The Repeated Prisoner’s Dilemma

$$\begin{gathered} 1-\delta \ 1 \ \delta^{T+1} \end{gathered} \sum_{i=0}^T \delta^t y_i\left(a^t\right) .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

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

## 经济代写|博弈论代写Game Theory代考|BUS-G303

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

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

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$\mathrm{D}$。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

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

## 经济代写|博弈论代写Game Theory代考|Econ411

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|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

$$S_i=\underset{h_i \in H_i}{\times} A\left(h_i\right) .$$

$$\prod_{h_i \in H_i} #\left(A\left(h_i\right)\right) \text {. }$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

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

## 经济代写|博弈论代写Game Theory代考|PSCI288

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

$$\mathrm{I}_{i 0} u_0(y(\theta), \theta) .$$

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

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

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

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

## 经济代写|博弈论代写Game Theory代考|CS4

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|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

$$q_2\left(\theta_2\right) \in \underset{4}{\arg \max }\left{q_2\left(\theta_2-q_1-q_2\right)\right} \Rightarrow q_2\left(\theta_2\right)=\left(\theta_2-q_1\right) / 2 .$$

\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}

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

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

## 经济代写|博弈论代写Game Theory代考|ECON314

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|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

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

$$1.5 y+3.5(1-y)>2 y+3(1-y),$$

$$y<\frac{1}{2} .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

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

## 经济代写|博弈论代写Game Theory代考|PSCI288

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|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|The Model

$$g_{\mathrm{i}}(a)=\sum_y \pi_y(a) r_i\left(a_i, y\right) .$$

$$h^{\prime}-\left(y^0, y^1, \ldots, y^2{ }^1\right) \text {. }$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

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

## 经济代写|博弈论代写Game Theory代考|ECN614

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|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

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

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