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# 数学代写|运筹学代写Operations Research代考|MATH3830 GAMES WITH SADDLE POINT

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## 数学代写|运筹学代写Operations Research代考|GAMES WITH SADDLE POINT

As illustrated in the above example

There are two players: companies A and B competing with each other to garner maximum sales by launching a new product at the earliest.

Both A and B have three strategies to select from the most appropriate one. Company A’s strategies are represented in different rows and of B’s in columns.

Row represents an evaluation of strategy by company B, which tells them what strategy company A would play to maximize gains. For instance, if company A decides to play A1, i.e. it intends to launch the product 4 months before the deadline, then $\mathrm{B}$ has the option to respond by playing either B1, B2 or B3. Positive value of payoffs indicates gain for company A and loss for company B. From all three strategies played by company B, it would always incur a loss. Therefore, to incur minimum loss, company B would adopt strategy B3 that causes it to lose two units. Company A gains the same amount. However, if company A decides to play A2 and launches the product before any of the dates by company B, then it would always gain irrespective of B’s strategy. B would opt for a strategy that would incur it minimum loss that is B3 with a loss of 1 unit. Finally, if A expedites the launch to 8 months before the deadline, then it has a chance of incurring losses because of quality problems and company B would take its benefit by playing B2. It results in a gain of 2 and loss of the same amount to A (minus sign indicate loss to $\mathrm{A}$ and gain to $\mathrm{B}$ as both players are diametrically opposite to each other). Thus, from each row, the minimum value is selected. As these values indicate gain by A from evaluations by $\mathrm{B}$, so $\mathrm{A}$ would play with the maximum value, implying maximum gain. Thus, A would play A1 and enjoys a gain of two units. This principle is called as minimax rule.

On the other hand, column indicates the selection of the most appropriate strategy by company B that incurs a minimum loss. For instance, if B decides to play $\mathrm{B} 1$, then $\mathrm{A}$ would respond by playing either $\mathrm{A} 1, \mathrm{~A} 2$ or $\mathrm{A} 3$. As its purpose is to maximize the gains, so it would select A3 with a gain of four units. In this case, A would always gain as its launch schedule is in all three strategies much earlier than 1 month before the strategy of B. Company B would like to play B2 because it intends to gain two units, but only if A plays A3. Company A, being a rational player, would, in response, play A1, which gives it a maximum gain of four units. Interestingly, when B launches 5 months before (B3), then A gains maximum even if it launches later than $\mathrm{B}$, i.e. by playing $\mathrm{A} 1$, A gains 2 , which is more than if it plays $\mathrm{A} 2$ or $\mathrm{A} 3$. Thus, while selecting a strategy for $\mathrm{B}$, identify the maximum value from each column. However, the purpose of B is to minimize losses, so out of identified maximum values, select the minimum value. Thus, B would play $\mathrm{B} 3$ and incur a minimum loss of two units. This principle is called as maximin rule.

## 数学代写|运筹学代写Operations Research代考|PRinCiPle of dominanCe

To identify the most appropriate strategy for each player, the payoff table can also be deduced by using the principle of dominance where in inferior strategies are removed in succession until only one choice is left. The principle can be applied for both players. As player in the row (player A) would use a strategy that maximizes its gain, so a strategy is said to be dominated by the second strategy if the payoff values in the second strategy are more or at least equal to those in the first strategy. Player A would always select second strategy, as it would give better gain than the first strategy.
On the other hand, considering player in column (player B) purpose of minimizing losses, a strategy is said to be dominated by the second strategy if payoff values in the second strategy are less or at least equal to corresponding payoff values in the first strategy. Player B would then select the second strategy, as it would result in fewer losses than the first strategy.

These rules are applied on game shown in Table $9.1$ that is reproduced as under and renamed as Table $9.3$.

For instance, if A1 is compared with A2, then the value of payoff cell $\mathrm{a}{11}$, i.e. 3 is greater than that of cell $\mathrm{a}{21}$, i.e. 2. So, A would always select $\mathrm{a}{11}$. Comparison of $\mathrm{a}{12}$ with $\mathrm{a}{22}$ also results in selection of $\mathrm{a}{12}$ as payoff of 4 is greater than 3. Similarly, payoff in $\mathrm{a}{13}$ of 2 is greater than payoff in $\mathrm{a}{23}$ of 1 . This would allow company $\mathrm{A}$ to always play A1 in comparison with A2. A1 is called as a dominating strategy and A2 as dominated. This is shown in Table 9.4.

In the case of company B, comparing B1 with B3 would not give a clear dominating strategy. As payoff in cell $\mathrm{a}{11}$ of 3 is less than that of 4 in $\mathrm{a}{12}$, so $\mathrm{B}$ would always select $\mathrm{a}{11}$ indicating strategy $\mathrm{B} 1$. However, when the payoff of 4 in $\mathrm{a}{31}$ is compared with that of $-2$ in $\mathrm{a}{32}, \mathrm{~B}$ will select $\mathrm{a}{32}$ indicating strategy $\mathrm{B} 2$. Comparison of $\mathrm{B} 2$ and B3 also gives the same contradicting results. Finally, comparing B1 with B3 provides B3 as a dominating strategy as 2 is less than 3 and $-1$ is less than 4. So, B would always select B3 when compared with B1. This is shown in Table 9.5.

## 数学代写|运筹学代写Operations Research代考|GAMES WITH SADDLE POINT

A 和 B 都有三种策略，可以从中选择最合适的一种。公司 A 的战略在不同的行中表示，而 B 公司的战略在列中表示。

## 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。