Posted on Categories:Discrete Mathematics, 数学代写, 离散数学

# 数学代写|离散数学代写Discrete Mathematics代考|Congruous Sets

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|离散数学代写Discrete Mathematics代考|Congruous Sets

Remember that the Nash equilibrium concept represents the extreme version of congruity in which the players coordinate on a single strategy profile. In some settings, it may not be reasonable to expect such an extreme form of coordination. One reason is that there may not be a social institution that serves to coordinate beliefs and behavior. Another reason is that coordination on a single strategy profile may be inconsistent with best-response behavior in some games.
For an interesting example, consider the game shown in Figure 9.3. Suppose the players can communicate before the game to discuss how to coordinate their play. Would they coordinate on the Nash equilibrium strategy profile $(\mathrm{z}, \mathrm{m})$ ? Perhaps, but it would be a shame, for the players would get higher payoffs if they could coordinate on not playing strategies $\mathrm{z}$ and $\mathrm{m}$. Unfortunately, this kind of coordination cannot be captured by the equilibrium notion, as $(\mathrm{z}, \mathrm{m})$ is the only Nash equilibrium of the game.
One can define a more general notion of congruity that lies between rationalizability and Nash equilibrium, in which strategic uncertainty is reduced but not always eliminated. The key is to associate the congruity idea with sets of strategy profiles. For instance, for the game shown in Figure 9.3, consider the set of strategy profiles $X \equiv{\mathrm{w}, \mathrm{y}} \times{\mathrm{k}, 1}$. Notice that if player 1 is convinced that player 2 will select either $\mathrm{k}$ or 1 (but not $\mathrm{m}$ ), then player 1’s best response must be $\mathrm{w}$ or $\mathrm{y}$. Likewise, if player 2 thinks player 1 will select either w or $\mathrm{y}$, then player 2’s best responses are only strategies $\mathrm{k}$ and 1 . We can say that the set $X$ is a congruous set because coordinating on $X$ is consistent with common knowledge of best-response behavior. Here is a precise and general definition:
Consider a set of strategy profiles $X=X_1 \times X_2 \times \cdots \times X_n$, where $X_i \subset S_i$ for each player $i$. The set $X$ is called congruous if, for each player $i$, a strategy $s_i$ is included in $X_i$ if and only if there is a belief $\theta_{-i} \in \Delta X_{-i}$ (putting probability only on strategies in $X_{-i}$ ) such that $s_i \in B R_i\left(\theta_{-i}\right)$. The set $X$ is called weakly congruous if, for each player $i$ and each strategy $s_i \in X_i$, there is a belief $\theta_{-i} \in \Delta X_{-i}$ such that $s_i \in B R_i\left(\theta_{-i}\right)$.

## 数学代写|离散数学代写Discrete Mathematics代考|Aside: Experimental Game Theory

At this point in our tour of game theory, it is worthwhile to pause and reflect on the purpose and practicality of the theory. As I have already emphasized (and will continue to emphasize) in this book, game theory helps us to organize our thinking about strategic situations. It provides discipline for our analysis of the relation between the outcome of strategic interaction and our underlying assumptions about technology and behavior. Furthermore, the theory gives us tools for prescribing how people ought to behave-or, at least, what things people ought to consider-in strategic settings.
You might start to ask, however, whether the theory accurately describes and predicts real behavior. The answer is not so straightforward. There are two ways of evaluating whether game theory is successful in this regard. First, you might gather data about how people behave in real strategic situations. For example, you can observe where competing firms locate in a city, how team members interact within a firm, how managers contract with workers, and so forth. Then you can construct game-theoretic models in an attempt to make sense of the data. You can even perform statistical tests of the models. In fact, many empirical economists dedicate themselves to this line of work. These economists are constantly challenged by how to reconcile the complexities of the real world with necessarily abstract and unadorned theoretical models.
The second way of evaluating game theory’s predictive power is to bring the real world closer to the simple models. You can, for example, run laboratory experiments in which subjects are asked to play some simple matrix games. In fact, this sort of research-which is called experimental game theory-has become a little industry in itself. In many universities throughout the world, experimental economists herd students into laboratories that are filled with computer stations, attracting the students with the prospect of winning significant amounts of money. In comparison with experimental work done by researchers in other disciplines, the economists certainly have gotten one thing right: they pay well. By paying the subjects according to their performance in games, experimenters give them a strong incentive to think about how best to play.

## 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。