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# 经济代写|博弈论代考Game theory代写|E-327

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## 经济代写|博弈论代考Game theory代写|Renegotiation-Proofness in Infinitely Repeated Games

Pareto perfection and recursive efficiency for finite-horizon games are both defined using backward recursion from the terminal date. Defining renegotiation or Pareto perfection for infinite-horizon games has proved to be much more difficult, and there are currently many competing definitions. One of the earliest treatments is by Farrell and Maskin (1989), who define “weak renegotiation-proofness” for infinitely repeated games. This concept extends the “bygones are bygones” flavor of Pareto perfection by requiring that the set of renegotiation-proof equilibria at date $t$ be independent not only of the history $h^t$ but also of calendar time $t$. Weak renegotiation-proofness begins with the point of view that there is an exogenously chosen set of possible equilibrium payoffs $Q$ that is conceivable at any $t$ and $h^{\prime}$, and that each payoff in $Q$ must require only continuation payoffs corresponding to other equilibria in $Q$. Formally, let $c\left(\sigma ; h^{\prime}\right)$ be the continuation payoffs implied by $\sigma$ given history $h^{\prime}$, and let $C(\sigma)=$ $U_{1, h^{\prime}}\left(\sigma ; h^t\right)$ be the set of all continuation payoffs for strategy profile $\sigma$. Then, if $r \in Q$, there must be a perfect equilibrium $\sigma$ with payoffs $v$ such that $C(\sigma) \subseteq Q$. The set $Q$ is said to be weakly renegotiation-proof (WRP) if no equilibrium payoff in $Q$ is Pareto dominated by the payoffs of another ecquilibrium in $Q$.

This definition assigns a great deal of weight to the exogenous set of “social norms” $Q$. This allows, for example, any static equilibrium to be weakly renegotiation-proof as a one-point set. However, in the prisoner’s dilemma, the “grim” strategies of initial cooperation followed by the static equilibrium forever if someone deviates are not weakly renegotiationproof, as the payoffs corresponding to the “cooperative phase” of the strategies Pareto dominate those of the punishment phase. That is, once the payoffs of “always cooperate” are included in the set $Q$ of possible “agrecments,” the players will always renegotiatc from the unending punishment back to the cooperative phasc. Moreover, the strategies “perfect tit for tat,” defined by “play $\mathrm{C}$ in the first period, and subsequently play $\mathrm{C}$ if last period’s outcome was (C, C) or (D, D); play D if last period’s outcome was (D, C) or (C, D),” are not WRP either, as in the period immediately following a unilateral deviation it would be more efficient to ignore the deviation and play (C, C). These strategies are, however, subgame perfect for discount factors near I with the usual payoffs, i.e., those given in figure 5.5

## 经济代写|博弈论代考Game theory代写|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 satisficd, 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 Green-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 pricc. 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.

# 博弈论代写

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

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## MATLAB代写

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