## 经济代写|博弈论代写Game Theory代考|ECON3301

2022年12月29日

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## 经济代写|博弈论代写Game Theory代考|IMPERFECT INFORMATION

In this section, we will focus on games with imperfect information. Recall, these occur when players do not know one or more actions taken hy the other players and so find themselves in an information set with more than one history.

Assuming that Suzette has already made her decisions about how to carry out the bidding process, Mark and Ben are the players in this game. The game is sequential in nature, and Ben has incomplete information about Mark’s action. To create a simple model, we assume Mark can choose one of three actions: not bid (No), bid low (Lo), or bid high (Hi). Ben only knows whether Mark bids or does not bid; if Mark does bid, Ben does not know whether he bid high or low. Since Ben is not interested in doing the work for a low price, we assume he chooses between two actions: not bid (No) or bid high (Hi). Assigning vNM utilities consistent with this discussion, we obtain the game tree in Figure $6.1$ and Table 6.1.

We can see from the game tree the non-terminal histories are partitioned into three information sets, listed in Table 6.2. Only Ben faces an information set, Ben2, with more than one node (non-terminal history). Here Ben does not know which sequence of actions has occurred and is making his decision whether to choose No or Hi without knowing whether Mark has bid Hi or Lo.

It is important to consider players’ overall strategies, rather than just individual actions. In FoodPro, one pure strategy for Ben is to always choose No. Another is to choose No when Mark Bids (at information set Ben2) and Hi if Mark does not bid (at information set Ben1). Mark has three pure strategies, corresponding to the actions at his only nonterminal history: No, Lo, or Hi. For strategic games, we allowed players to adopt mixed strategies. While we could allow players in extensive games to adopt such mixed strategies, it is typically sufficient and more natural to allow players to randomize action choices at each information set based on probability distributions. We call such strategies behavior strategies and formally define them below.

Definition 6.1.1. A pure strategy for player $i$ is a function $s_i$ which assigns to each of the player’s information sets a possible action. A behavior strategy for player $i$ is a function $s_i$ which assigns to each of the player’s information sets a probability distribution over possible actions. If $s$ is used to denote a pure or behavior strategy, we will use $s_i(I)$ or simply $s(I)$ to denote the action or probability distribution over actions chosen by player $i$ at the assigned information set $I$, and $s_i(a \mid I)$ or simply $s(a \mid I)$ will denote the probability that player $i$ will choose action $a$ at information set $I$.

## 经济代写|博弈论代写Game Theory代考|ROMANS AND GERMANS

In this section, we will construct a model for an unfortunately common scenario in human history, that of war. We’ll use a historical scenario, but our model and analysis can apply to many modern-day situations.

Barron [8] describes a method for modeling a fictitious battle in the Greco-Persian wars. We present an adaptation of his model for the Greek and Roman battle described above.
Using RE and GT to name the players representing the Roman Empire and the Germanic tribe, the extensive game tree is given in Figure 6.8. To more easily discern the reasoning behind the payoffs, a short description of each outcome and the corresponding payoffs are rank ordered in Tables $6.3$ and 6.4. Observe that payoffs were assigned so that positive numbers correspond to an overall “win” for that player and negative numbers correspond to an overall “loss” for that player, and the numbers are to be interpreted as vNM utilities. We encourage the reader to consider whether different numbers might be more realistic.

This model is different from the three games analyzed in the first section because both players lack information. This lack of information is again indicated by the dashed line boxes around three pairs of the decision nodes. Within each of these boxes, the player who is making the decision does not know which history has occurred. We have also labeled each decision point so that they can be easily referenced. For example, within the GT1 box, when the German tribe is making its decision whether to defend the forest or lake, it does not know the Roman Empire legion’s decision whether to advance through the forest or over the lake, however, at the GT2 node (which may also be considered as an information set containing just this node), the German tribe knows that they have defended an advance across the lake but the Roman empire legion has attacked the village after advancing through the forest.
The Warfare game has three subgames: First, the entire game has the empty history as its root. Second, the subgame with root (Forest, Lake) which corresponds in Figure $6.8$ to the node labeled GT2 and everything to the right of GT2. Third, the subgame with root (Lake, Forest) which corresponds in Figure $6.8$ to the node labeled GT3 and everything to the right of GT3. The effect of requiring the consistency condition to hold on every subgame rather than just the entire game is to ensure players choose an equilibrium even on subgames that have zero probability of being reached.

We now find the weak sequential equilibria. Figure $6.8$ includes variable names below edges and nodes corresponding to the probabilities associated with an arbitrary assessment. Assume that the assessment is a weak sequential equilibrium.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|ROMANS AND GERMANS

Barron [8] 描述了一种模拟希波战争中虚构战斗的方法。我们展示了他对上述希腊和罗马战争模型的改编。

Warfare 游戏分为三个子游戏：首先，整个游戏以空历史为根。二、图中对应的根为(Forest, Lake)的子博弈6.8到标记为 GT2 的节点以及 GT2 右侧的所有内容。三、图中对应的根为(Lake, Forest)的子博弈6.8到标记为 GT3 的节点以及 GT3 右侧的所有内容。要求一致性条件适用于每个子博弈而不仅仅是整个博弈的效果是确保玩家即使在达到零概率的子博弈中也选择均衡。

## 有限元方法代写

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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。