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# 经济代写| Decision trees and actions微观经济学代写

## 经济代写

2.1. Perfect information. Before defining a decision situation in extensive form, we need to clarify what we mean by a partition.

DEFINITION III.1 (PARTITION). Let $M$ be any nonempty set. A partition of $M$ is a subset $\mathcal{P}{M}=\left{M{1}, \ldots, M_{k}\right}$ of the power set $2^{M}$ such that
hold. $B y \mathcal{P}{M}(m)$ we mean the element of $\mathcal{P}{M}$ that contains $m \in M$. The elements of partitions are often called components. A component with one element only is called a singleton.

Most of the time, a partition will not contain the empty set but we allow for this possibility.

EXERCISE III.1. Write down two partitions of $M:={1,2,3}$. Find $\mathcal{P}_{M}$ (1) in each case.

We begin by describing a decision situation. (This description is not a very formal definition but leans on the investment-marketing decision situation.)

DEfINITION III.2 (EXTENSIVE-FORM DECISION SITUATION). A decision situation (in extensive form and for perfect information) $\Delta=\left(V, u,\left(A_{d}\right)_{d \in D}\right)$ is given by

• a tree with node set $V$ where the nodes are often denoted by $v_{0}$, $v_{1}, \ldots$ together with
• links that connect the nodes, directly or indirectly.
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III. DECISIONS IN EXTENSIVE FORM
• A tree has an initial node $v_{0}$ and for every node $v$ thene exists exactly one trail (consisting of links) from $v_{0}$ to $v$ (see below). Trails are indicated by brackets \langle\rangle that contain at least two nodes.
• A trail is called maximal if it cannot be extended.
• The length of a trail is defined in the obvious manner (just go from one node to successor nodes and count the number of steps). The length of a tree is defined by its longest trail.
• $D$ is the set of non-terminal nodes that are also called decision nodes. $A_{d}$ is the set of actions that can be chosen at decision node d. Every link at $d$ corresponds to exactly one action. The set of all actions is defined by $A=\cup_{d \in D} A_{d}$.
• $E$ is the set of end nodes where a payoff function $u: E \rightarrow \mathbb{R}$ records the payoffs.

We have three decision nodes and six actions in the investment-marketing decision situation. The link from $v_{1}$ to $v_{3}$ in figure III.1 corresponds to action $M$ if the firm has chosen I at the first stage. Action M after action nI is a different link (the one from $v_{2}$ to $v_{5}$ ). Furthermore, observe $D=\left{v_{0}, v_{1}, v_{2}\right}$, $E=\left{v_{3}, v_{4}, v_{5}, v_{6}\right}, A_{v_{2}}={\mathrm{M}, \mathrm{nM}}, A={\mathrm{I}, \mathrm{nI}, \mathrm{M}, \mathrm{nM}} .$ The trail $\left\langle v_{0}, v_{3}\right\rangle$ has length 2 , while the length of trail $\left\langle v_{1}, v_{3}\right\rangle$ is 1 .

EXERCISE III.2. What is the length of the investment-marketing tree above? How about the absent minded driver (where you disregand the dotted line)? Indicate all the maximal trails in these two decision situations.
2.2. Imperfect information. The above definition refers to “perfect information”. This means that the decision maker knows the decision node at which he finds himself. In contrast, under “imperfect information” the decision maker does not know exactly the current decision node. We represent imperfect information by information sets that gather decision nodes between which the decision maker cannot distinguish. Therefore, the actions available at different nodes in an information set have to be the same. The absent-minded driver provides an example.

DEFINITION III.3 (INFORMATION PARTITION). A decision situation (in extensive form and for imperfect information) $\Delta=\left(V, u, I,\left(A_{d}\right){d \in D}\right)$ equals the one for perfect information with the following exception: There exists a partition I (called information partition) of the decision nodes $D$. The elements of I are called information sets (which are components of I). The actions at decision nodes belonging to the same information set have to be identical: $A{a}=A_{d^{\prime}}$ for all $d, d^{\prime} \in I(d)$.

2.1。完善的信息。在以广泛的形式定义决策情况之前，我们需要澄清分区的含义。

• 具有节点集 $V$ 的树，其中节点通常由 $v_{0}$、$v_{1}、\ldots$ 和
• 直接或间接连接节点的链接。
此外：
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三、广泛形式的决定
• 一棵树有一个初始节点 $v_{0}$ 并且对于每个节点 $v$ 都存在从 $v_{0}$ 到 $v$ 的一条路径（由链接组成）（见下文）。路径由包含至少两个节点的括号 \langle\rangle 指示。
• 如果一条路径不能扩展，则称为最大路径。
• 路径的长度以显而易见的方式定义（只需从一个节点到后继节点并计算步数）。树的长度由其最长的路径定义。
• $D$ 是非终端节点的集合，也称为决策节点。 $A_{d}$ 是可以在决策节点 d 处选择的动作集。 $d$ 处的每个链接都对应一个操作。所有动作的集合由 $A=\cup_{d \in D} A_{d}$ 定义。
• $E$ 是支付函数 $u: E \rightarrow \mathbb{R}$ 记录支付的端节点集。

2.2.信息不完善。上述定义指的是“完美信息”。这意味着决策者知道他所在的决策节点。相反，在“不完全信息”下，决策者并不确切知道当前的决策节点。我们通过收集决策者无法区分的决策节点的信息集来表示不完全信息。因此，信息集中不同节点可用的动作必须相同。心不在焉的司机提供了一个例子。

## 经济代考

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## 编码理论代写

1. 数据压缩（或信源编码
2. 前向错误更正（或信道编码
3. 加密编码
4. 线路码

## 复分析代考

(1) 提到复变函数 ，首先需要了解复数的基本性左和四则运算规则。怎么样计算复数的平方根， 极坐标与 $x y$ 坐标的转换，复数的模之类的。这些在高中的时候囸本上都会学过。
(2) 复变函数自然是在复平面上来研究问题，此时数学分析里面的求导数之尖的运算就会很自然的 引入到复平面里面，从而引出解析函数的定义。那/研究解析函数的性贡就是关楗所在。最关键的 地方就是所谓的Cauchy一Riemann公式，这个是判断一个函数是否是解析函数的关键所在。
(3) 明白解析函数的定义以及性质之后，就会把数学分析里面的曲线积分 $a$ 的概念引入复分析中， 定义几乎是一致的。在引入了闭曲线和曲线积分之后，就会有出现复分析中的重要的定理: Cauchy 积分公式。 这个是易分析的第一个重要定理。