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数学代写|离散数学代写Discrete Mathematics代考|Randomization in Sports

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数学代写|离散数学代写Discrete Mathematics代考|Randomization in Sports

For another example, take the tennis-service game of Chapter 7’s Guided Exercise, whose payoff matrix is reproduced in Figure 11.2. Recall that each player’s strategy $\mathrm{F}$ is removed in the iterated-dominance procedure, so the set of rationalizable strategies for each player is ${\mathrm{C}, \mathrm{B}}$. The game has no Nash equilibrium in pure strategies. In any mixed-strategy equilibrium, the players will put positive probability on only rationalizable strategies. Thus, we know a mixedstrategy equilibrium will specify a strategy $(0, p, 1-p)$ for player 1 and a strategy $(0, q, 1-q)$ for player 2 . In this strategy profile, $p$ is the probability that player 1 selects C, and $1-p$ is the probability that he selects B; likewise, $q$ is the probability that player 2 selects $\mathrm{C}$, and $1-q$ is the probability that she selects B.
To calculate the mixed-strategy equilibrium in the tennis example, observe that against player 2’s mixed strategy, player 1 would get an expected payoff of
$$q \cdot 0+(1-q) \cdot 3=3-3 q$$
if he selects $\mathrm{C}$; whereas by choosing $\mathrm{B}$, he would expect
$$q \cdot 3+(1-q) \cdot 2=2+q$$

数学代写|离散数学代写Discrete Mathematics代考|TECHNICAL NOTES

The following summarizes the steps required to calculate mixed-strategy Nash equilibria for simple two-player games.
Procedure for finding mixed-strategy equilibria:

1. Calculate the set of rationalizable strategies by performing the iterateddominance procedure.
1. Restricting attention to rationalizable strategies, write equations for each player to characterize mixing probabilities that make the other player indifferent between the relevant pure strategies.
2. Solve these equations to determine equilibrium mixing probabilities.

If each player has exactly two rationalizable strategies, this procedure is quite straightforward. If a player has more than two rationalizable strategies, then there are several cases to consider; the various cases amount to trying different combinations of pure strategies over which the players may randomize. For example, suppose that $\mathrm{A}, \mathrm{B}$, and $\mathrm{C}$ are all rationalizable for a particular player. Then, in a mixed-strategy equilibrium, it may be that this player mixes between $\mathrm{A}$ and $\mathrm{B}$ (putting zero probability on $\mathrm{C}$ ), mixes between $\mathrm{A}$ and $\mathrm{C}$ (putting zero probability on $B$ ), mixes between $\mathrm{B}$ and $\mathrm{C}$ (putting zero probability on $\mathrm{A}$ ), or mixes between $\mathrm{A}, \mathrm{B}$, and C. There are also cases in which only one of the players mixes.

Note that every pure-strategy equilibrium can also be considered a mixedstrategy equilibrium-where all probability is put on one pure strategy. All of the games analyzed thus far have at least one equilibrium (in pure or mixed strategies). In fact, this is a general theorem. ${ }^4$
Result: Every finite game (having a finite number of players and a finite strategy space) has at least one Nash equilibrium in pure or mixed strategies.

数学代写|离散数学代写Discrete Mathematics代考|Randomization in Sports

$$q \cdot 0+(1-q) \cdot 3=3-3 q$$

$$q \cdot 3+(1-q) \cdot 2=2+q$$

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。