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# 经济代写|博弈论代考Game theory代写|Critiques of Backward Induction

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## 经济代写|博弈论代考Game theory代写|Critiques 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 $\operatorname{gets} 2$.

Since only one 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 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 cither “mistakes” by future players or the possibility that player 3 might intentionally play $D$.

## 经济代写|博弈论代考Game theory代写|Critiques of Subgame Perfection

Since subgame perfection is an extension of backward induction, it is vulnerable to the critiques just discussed. Moreover, subgame perfection requires that players all agree on the play in a subgame even if that play cannot be predicted from backward-induction arguments. This point is emphasized by Rabin (1988), who proposes alternative, weaker equilibrium refinements that allow players to disagree about which Nash equilibrium will occur in a subgame off the equilibrium path.

To see the difference this makes, consider the following three-player game. In the first stage, player 1 can either play $L$, ending the game with payoffs $(6,0,6)$, or play $R$, which gives the move to player 2 . Player 2 can then either play $R$, ending the game with payoffs $(8,6,8)$, or play $L$, in which case players 1 and 3 (but not player 2) play a simultaneous-move “coordination game” in which they each choose F or $G$. If their choices differ, they each receive 7 and player 2 gets 10 ; if the choices match, all three players receive 0 . This game is depicted in figure 3.20 .

The coordination game between players 1 and 3 at the third stage has three Nash equilibria: two in pure strategies with payoffs $(7,10,7)$ and a mixed-strategy equilibrium with payoffs $\left(3 \frac{1}{2}, 5,3 \frac{1}{2}\right)$. If we specify an equilibrium in which players 1 and 3 successfully coordinate, then player 2 plays L and so player 1 plays $R$, expecting a payoff of 7 . If we specify the inefficient mixed equilibrium in the third stage, then player 2 will play $R$ and again player 1 plays $R$, this time expecting a payoff of 8 . Thus, in all subgame-perfect equilibria of this game, player 1 plays $R$.

As Rabin argues, it may nevertheless be reasonable for player 1 to play $L$. He would do so if he saw no way to coordinate in the third stage, and hence expected a payoff of $3 \frac{1}{2}$ conditional on that stage being reached, but feared that player 2 would believe that play in the third stage would result in coordination on an efficient equilibrium.

The point is that subgame perfection supposes not only that the players expect Nash equilibria in all subgames but also that all players expect the same equilibria. Whether this is plausible depends on the reason one thinks an equilibrium might arise in the first place.

# 博弈论代写

## 经济代写|博弈论代考Game theory代写|Critiques of Subgame Perfection

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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。