## 经济代写|博弈论代写Game Theory代考|ECON40010

statistics-lab™ 为您的留学生涯保驾护航 在代写博弈论Game Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写博弈论Game Theory代写方面经验极为丰富，各种代写博弈论Game Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|博弈论代写Game Theory代考|Nim, Poker Nim, and the Mex Rule

In this section, we prove a sweeping statement (Theorem 1.14): Every impartial game is equivalent to some Nim heap. Note that this means a single Nim heap, even if the game itself is rather complicated, for example a general Nim position, which is a game sum of several Nim heaps. This can proved without any reference to playing Nim optimally. As we will see at the end of this section with the help of Figure 1.2, the so-called mex rule that underlies Theorem $1.14$ can be used to discover the role of powers of two for playing sums of Nim heaps.

We use the following notation for Nim heaps. If $G$ is a single Nim heap with $n$ tokens, $n \geq 0$, then we denote this game by $* n$. This game is completely specified by its $n$ options, and they are defined recursively as follows:
options of $* n: \quad * 0, * 1, * 2, \ldots, (n-1)$. Note that $0$ is the empty heap with no tokens, that is, $* 0=0$; we will normally continue to just write 0 .

We can use (1.18) as the definition of $* n$. For example, the game $* 4$ is defined by its options $* 0, * 1, * 2, * 3$. It is very important to include $* 0$ in that list of options, because it means that $* 4$ has a winning move. Condition (1.18) is a recursive definition of the game $* n$, because its options are also defined by reference to such games $* k$, for numbers $k$ smaller than $n$. This game fulfills the ending condition because the heap gets successively smaller in any sequence of moves.

A general Nim position is a game sum of several Nim heaps. Earlier we had written such a position by just listing the sizes of the Nim heaps, such as $1,2,3$ in (1.1). The fancy way to write this is now $* 1+* 2+* 3$, a sum of games.

The game of Poker Nim is a variation of Nim. Suppose that each player is given, at the beginning of the game, some extra “reserve” tokens. Like Nim, the game is played with heaps of tokens. In a move, a player can choose, as in ordinary Nim, a heap and remove some tokens, which he can add to his reserve tokens. A second, new kind of move is to add some of the player’s reserve tokens to some heap (or even to create an entire new heap with these tokens). These two kinds of moves are the only ones allowed.

## 经济代写|博弈论代写Game Theory代考|Sums of Nim Heaps

In this section, we derive how to compute the Nim value for a general Nim position, which is a sum of different Nim heaps. This will be the Nim sum that we have defined using the binary representation, now cast in the language of game sums and equivalent games, and without assuming the binary representation.

For example, we know that $* 1+* 2+* 3 \equiv 0$, so by Lemma $1.12, * 1+* 2$ is equivalent to $* 3$. In general, however, the sizes of the Nim heaps cannot simply be added to obtain the equivalent Nim heap, because $* 2+* 3$ is also equivalent to $* 1$, and $* 1+* 3$ is equivalent to $* 2$.

If $* k \equiv * n+* m$, then we call $k$ the $\operatorname{Nim}$ sum of $n$ and $m$, written $k=n \oplus m$. The following theorem states that the Nim sum of distinct powers of two is their arithmetic sum. For example, $1=2^0$ and $2=2^1$, so $1 \oplus 2=1+2=3$.

Theorem 1.15. Let $n \geq 1$, and $n=2^a+2^b+2^c+\cdots$, where $a>b>c>\cdots \geq 0$. Then
$$• n \equiv \left(2^a\right)+\left(2^b\right)+*\left(2^c\right)+\cdots \text {. }$$
We first discuss the implications of this theorem, and then prove it. The expression $n=2^a+2^b+2^c+\cdots$ is an arithmetic sum of distinct powers of two. Any $n$ is uniquely given as such a sum. It amounts to the binary representation of $n$, which, if $n<2^{a+1}$, gives $n$ as the sum of all powers $2^a, 2^{a-1}, 2^{a-2}, \ldots, 2^0$ where each power of two is multiplied with 0 or 1 , the binary digit for the respective position. For example,
$$9=8+1=1 \cdot 2^3+0 \cdot 2^2+0 \cdot 2^1+1 \cdot 2^0,$$
so that 9 in decimal is written as 1001 in binary. Theorem $1.15$ uses only the distinct powers of two $2^a, 2^b, 2^c, \ldots$ that correspond to the digits 1 in the binary representation of $n$.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|Nim, Poker Nim, and the Mex Rule

Poker Nim 游戏是 Nim 的变体。假设在游戏开始时给每个玩家一些额外的“保留”标记。与 Nim 一样，该游戏是 用大量代币玩的。在移动中，玩家可以像在普通 Nim 中一样选择一个堆并移除一些令牌，他可以将这些令牌添 加到他的储备令牌中。第二种新的移动方式是将玩家的一些储备令牌添加到一些堆中（或者甚至用这些令牌创建 一个全新的堆）。这两种动作是唯一允许的。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 经济代写|博弈论代写Game Theory代考|ECOS3012

statistics-lab™ 为您的留学生涯保驾护航 在代写博弈论Game Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写博弈论Game Theory代写方面经验极为丰富，各种代写博弈论Game Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|博弈论代写Game Theory代考|Top-down Induction

When talking about combinatorial games, we will often use for brevity the word game for “game position”. Every game $G$ has finitely many options $G_1, \ldots, G_m$ that are reached from $G$ by one of the allowed moves in $G$, as in this picture:

If $m=0$ then $G$ has no options. We denote the game with no options by 0 , which by the normal play convention is a losing game. Otherwise the options of $G$ are themselves games, defined by their respective options according to the rules of the game. In that way, any game is completely defined by its options. In short, the starting position defines the game completely.

We introduce a certain type of mathematical induction for games, which is applied to a partial order (see the background material text box on the next page).
Consider a set $S$ of games, defined, for example, by a starting game and all the games that can reached from it via any sequence of moves of the players. For two games $G$ and $H$ in $S$, call $H$ simpler than $G$ if there is a sequence of moves that leads from $G$ to $H$. We allow $G=H$ where this sequence is empty. The relation of being “simpler than” defines a partial order which for the moment we denote by $\leq$. Note that $\leq$ is antisymmetric because it is not possible to reach $G$ from $G$ by a nonempty sequence of moves because this would violate the ending condition. The ending condition for games implies the following property:
Every nonempty subset of $S$ has a minimal element.
If there was a nonempty subset $T$ of $S$ without a minimal element, then we could produce an infinite play as follows: Start with some $G$ in $T$. Because $G$ is not minimal, there is some $H$ in $T$ with $H<G$, so there is some sequence of moves from $G$ to $H$. Similarly, $H$ is not minimal, so another game in $T$ is reached from $H$. Continuing in this manner creates an infinite sequence of moves, which contradicts the ending condition.

## 经济代写|博弈论代写Game Theory代考|Game Sums and Equivalence of Games

Combinatorial games often “decompose” into parts in which players can move independently, and the players then have to decide in which part to make their move. This is captured by the important concept of a sum of games.

Definition 1.4. Suppose that $G$ and $H$ are game positions with options (positions reached by one move) $G_1, \ldots, G_k$ and $H_1, \ldots, H_m$, respectively. Then the options of the game sum $G+H$ are
$$G_1+H, \ldots, G_k+H, \quad G+H_1, \ldots, G+H_m .$$
The first list of options $G_1+H, \ldots, G_k+H$ in (1.11) simply means that the player makes his move in $G$, the second list $G+H_1, \ldots, G+H_m$ that he makes his move in $H$; the other part of the game sum remains untouched. As an example, a Nim position is simply the game sum of its individual Nim heaps, because the player moves in exactly one of the heaps.

Definition $1.4$ is a recursive definition, because the game sum is defined in terms of its options, which are themselves game sums (but they are simpler games).
The sum of games turns out to define an abelian group on the (appropriately defined) set of games. It is a commutative and associative operation: for any games $\mathrm{G}, H, J$,
$$G+H=H+G \text { and }(G+H)+J=G+(H+J) \text {. }$$
The first condition (commutativity) holds because the order of the options of a game, used in (1.11), does not matter. The second condition (associativity) holds because both $(G+H)+J$ and $G+(H+J)$ mean in effect that the player decides to move in $G$, in $H$, or in $J$, leaving the other two parts of the game sum unchanged. We can therefore assume the equalities (1.12). More generally, in a sum of several games $G_1, \ldots, G_n$ the player moves in exactly one of these games, which does not depend on how these games are arranged, so that we can write this sum unambiguously without parentheses as $G_1+\cdots+G_n$.

The losing game 0 which has no options is a zero (neutral element) for game sums: It fulfills $G+0=G$ for any game $G$, because the game 0 is “invisible” when added to $G$.

In order to obtain a group, every game $G$ needs to have a “negative” game $-G$ so that $G+(-G)=0$. However, this equality cannot hold as stated as soon as the game $G$ has options, because then the game $G+(-G)$ also has options but 0 has none. Instead, we need a more general condition
$$G+(-G) \equiv 0,$$
where $G \equiv H$ means that the two games $G$ and $H$ are equivalent, according to the following definition.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|Top-down Induction

: $S$ 有一个最小的元素。

## 经济代写|博弈论代写Game Theory代考|Game Sums and Equivalence of Games

$$G_1+H, \ldots, G_k+H, \quad G+H_1, \ldots, G+H_m .$$

$$G+H=H+G \text { and }(G+H)+J=G+(H+J) .$$

$$G+(-G) \equiv 0,$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 经济代写|博弈论代写Game Theory代考|ECON6025

statistics-lab™ 为您的留学生涯保驾护航 在代写博弈论Game Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写博弈论Game Theory代写方面经验极为丰富，各种代写博弈论Game Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|博弈论代写Game Theory代考|Nim and Combinatorial Games

Combinatorial game theory is about perfect-information two-player games, such as Checkers, Go, Chess, or Nim, which are analyzed using their rules. It tries to answer who will win in a game position (assuming optimal play on both sides), and to quantify who is ahead and by how much. The topic has a rich mathematical theory that relates to discrete mathematics, algebra, and (not touched here) computational complexity, and highly original ideas specific to these games.
Combinatorial games are not part of “classical” game theory as used in economics. However, they nicely demonstrate that game theory is about rigorous, and often unfamiliar, mathematical concepts rather than complex techniques.
This chapter is only an introduction to combinatorial games. It presents the theory of impartial games where in any game position both players have the same allowed moves. We show the powerful and surprisingly simple result (Theorem 1.14), independently found by Sprague (1935) and Grundy (1939), that every impartial game is equivalent to a “Nim heap” of suitable size.

In Section $1.8$ we give a short glimpse into the more general theory of partizan games, where the allowed moves may depend on the player (e.g., one player can move the white pieces on the game board and the other player the black pieces).
For a deeper treatment, the final Section $1.9$ of this chapter lists some excellent textbooks on combinatorial games. They treat impartial games as a special case of general combinatorial games. In contrast, we first treat the simpler impartial games in full.

## 经济代写|博弈论代写Game Theory代考|Prerequisites and Learning Outcomes

Combinatorial games are two-player win-lose games of perfect information, that is, every player is perfectly informed about the state of play (unlike, for example, the card games Bridge or Poker that have hidden information). The games do not have chance moves like rolling dice or shuffling cards. When playing the game, the two players always alternate in making a move. Every play of the game ends with a win for one player and a loss for the other player (some games like Chess allow for a draw as an outcome, but not the games we consider here).

The game has a (typically finite) number of positions, with well-defined rules that define the allowed moves to reach the next position. The rules are such that play will always come to an end because some player is unable to move. This is called the ending condition. We assume the normal play convention that a player unable to move loses. The alternative to normal play is misère play, where a player who is unable to move wins (so the previous player who has made the last move loses).

We study impartial games where the available moves in a game position do not depend on whose turn it is to move. If that is not the case, as in Chess where one player can only move the white pieces and the other player the black pieces, the game is called partizan.

For impartial games, the game Nim plays a central role. A game position in Nim is given by some heaps of tokens, and a move is to remove some (at least one, possibly all) tokens from one of the heaps. The last player able to move wins the game, according to the normal play convention.

We analyze the Nim position 1,2,3, which means three heaps with one, two, and three tokens, respectively. One possible move is to remove two tokens from the heap of size three, like here: which we write as a move from $1,2,3$ to $1,2,1$. Because the move can be made in any one heap, the order of the heap sizes does not matter, so the position $1,2,1$ could also be written as $1,1,2$. The options of a game position are the positions that can be reached by a single legal move (according to the game rules) from the player to move. We draw them with moves shown as downward lines, like here,

where the first option 2,3 is obtained by removing from $1,2,3$ the entire heap of size 1 , the second option 1,1,3 by removing one token from the heap of size 2 , and so on. The game tree is obtained by continuing to draw all possible moves in this way until play ends (game trees are studied in much more detail in Chapter 4). We may conflate options with obvious equal meaning, such as the positions 1,1,2 and 1,2,1 that can be reached from 1,2,2. However, we do not draw moves to the same position from two different predecessors, such as 1,1,2 that can be reached from $1,1,3$ and 1, 2,2. Instead, such a position like 1,1,2 will be repeated in the game tree, so that every position has a unique history of moves.

In an impartial game, the available moves in a game position are by definition independent of the player to move. A game position belongs therefore to exactly one of two possible outcome classes, namely it is either a winning or a losing position. “Winning” or “losing” applies to the player whose turn it is to move, assuming optimal play. A winning position means that the player can force a win with a suitable first “winning” move (and subsequent winning moves at all later positions). A losing position means that every move from the current position leads to a winning position of the other player, who can then force a win, so that the current player will lose.

# 博弈论代考

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 经济代写|博弈论代写Game Theory代考|ECON3503

statistics-lab™ 为您的留学生涯保驾护航 在代写博弈论Game Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写博弈论Game Theory代写方面经验极为丰富，各种代写博弈论Game Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|博弈论代写Game Theory代考|Probabilities, information and entropy

Consider $n$ mutually exclusive events $E_1, \ldots, E_n$, and expect that any one of these, say $E_i$, indeed occurs “with probability” $p_i=\operatorname{Pr}\left(E_i\right)$. Then the parameters $p_i$ form a probability distribution $p \in \mathbb{R}^{\mathcal{E}}$ on the set $\mathcal{E}=\left{E_1, \ldots, E_n\right}$, i.e., the $p_i$ are nonnegative real numbers that sum up to 1 :
$$p_1+\cdots+p_n=1 \quad \text { and } \quad p_1, \ldots, p_n \geq 0 .$$
If we have furthermore a measuring or observation device $f$ that produces the number $f_i$ if $E_i$ occurs, then these numbers have the expected value
$$\mu(f)=f_1 p_1+\cdots+f_n p_n=\sum_{k=1}^n f_i p_i=\langle f \mid p\rangle .$$
In a game-theoretic context, a probability is often a subjective evaluation of the likelihood for an event to occur. The gambler, investor, or general player may not know in advance what the future will bring, but has more or less educated guesses on the likelihood of certain events. There is a close connection with the notion of information.

Intensity. We think of the intensity of an event $E$ as a numerical parameter that is inversely proportional to its probability $p=\operatorname{Pr}(E)$ with which we expect its occurrence to be: the smaller $p$, the more intensely felt is an actual occurrence of $E$. For simplicity, let us take $1 / p$ as our objective intensity measure.

Remark $1.7$ (Fechner’s law). According to Fechner, ${ }^{11}$ the intensity of a physical stimulation is physiologically felt on a logarithmic scale. Well-known examples are the Richter scale for earthquakes or the decibel scale for the sound.

Following FECHNER, we feel the intensity of an event $E$ that we expect with probability $p$ on a logarithmic scale and hence according to a function of type
$$I_a(p)=\log _a(1 / p)=-\log _a p,$$
where $\log _a p$ is the logarithm of $p$ relative to the basis $a>0$ (see Ex. 1.7). In particular, the occurrence of an “impossible” event, which we expect with zero probability, has infinite intensity
$$I_a(0)=-\log _a 0=+\infty .$$

## 经济代写|博弈论代写Game Theory代考|Systems

A system is a physical, economic, or other entity that is in a certain state at any given moment. Denoting by $\mathfrak{S}$ the collection of all possible states $\sigma$, we identify the system with $\mathfrak{S}$. This is, of course, a very abstract definition. In practice, one will have to describe the system states in a way that is suitable for a concrete mathematical analysis. To get a first idea of what is meant, let us look at some examples.

Chess. A system arises from a game of chess as follows: A state of chess is a particular configuration $C$ of the chess pieces on the chess board, together with the information which of the two players ( ” $B$ ” or ” $W$ “) is to draw next. If $\mathfrak{C}$ is the collection of all possible chess configurations, a state could thus be described as a pair
$$\sigma=(C, p) \quad \text { with } C \in \mathfrak{C} \text { and } p \in{B, W} .$$
In a similar way, a card game takes place in the context of a system whose states are the possible distributions of cards among the players together with the information which players are to move next.

Economies. The model of an exchange economy involves a set $N$ of agents and a set $\mathcal{G}$ of certain specified goods. A bundle for agent $i \in N$ is a data vector
$$b=\left(b_G \mid G \in \mathcal{G}\right) \in \mathbb{R}^{\mathcal{G}},$$
where the component $b_G$ indicates that the bundle $b$ comprises $b_G$ units of the good $G \in \mathcal{G}$. Denoting by $\mathcal{B}$ the set of all possible bundles, we can describe a state of the exchange economy by a data vector
$$\beta=\left(\beta_i \mid i \in N\right) \in \mathcal{B}^N$$
that specifies each agent $i$ ‘s particular bundle $\beta_i \in \mathcal{B}$.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|Probabilities, information and entropy

$$p_1+\cdots+p_n=1 \quad \text { and } \quad p_1, \ldots, p_n \geq 0 .$$

$$\mu(f)=f_1 p_1+\cdots+f_n p_n=\sum_{k=1}^n f_i p_i=\langle f \mid p\rangle .$$

$$I_a(p)=\log _a(1 / p)=-\log _a p,$$

$$I_a(0)=-\log _a 0=+\infty$$

## 经济代写|博弈论代写Game Theory代考|Systems

$$\sigma=(C, p) \quad \text { with } C \in \mathfrak{C} \text { and } p \in B, W .$$

$$b=\left(b_G \mid G \in \mathcal{G}\right) \in \mathbb{R}^{\mathcal{G}},$$

$$\beta=\left(\beta_i \mid i \in N\right) \in \mathcal{B}^N$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 经济代写|博弈论代写Game Theory代考|ECON90022

statistics-lab™ 为您的留学生涯保驾护航 在代写博弈论Game Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写博弈论Game Theory代写方面经验极为丰富，各种代写博弈论Game Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|博弈论代写Game Theory代考|Algebra of functions and matrices

While the coefficients of data vectors or matrices can be quite varied (colors, sounds, configurations in games, etc.), we will typically deal with numerical data so that coordinate vectors have real numbers as their component values. Hence we deal with coordinate spaces of the type
$$\mathbb{R}^S={f: S \rightarrow \mathbb{R}}$$
Addition and scalar multiplication. The sum $f+g$ of two coordinate vectors $f, g \in \mathbb{R}^S$ is the vector of component sums $(f+g)_s=f_s+g_s$, i.e.
$$f+g=\left(f_s+g_s \mid s \in S\right) .$$
For any scalar $\lambda \in \mathbb{R}$, the scalar product $\lambda f$ multiplies each component of $f \in \mathbb{R}^S$ by $\lambda$ :
$$\lambda f=\left(\lambda f_s \mid s \in S\right) .$$
Warning. There are many – quite different – notions for “multiplication” operations with vectors.

Products. The (function) product $f \bullet g$ of two vectors $f, g \in \mathbb{R}^S$ is the vector with the componentwise products, i.e.,
$$f \bullet g=\left(f_s g_s \mid s \in S\right) .$$
In the special case of matrices $A, B \in \mathbb{R}^{X \times Y}$ the function product of $A$ and $B$ is called the HadamarD ${ }^5$ product
$$A \bullet B \in \mathbb{R}^{X \times Y} \quad\left(\text { with coefficients }(A \bullet B){x y}=A{x y} B_{x y}\right) .$$
Warning. The Hadamard product is quite different than the standard matrix multiplication rule (3) below.

## 经济代写|博弈论代写Game Theory代考|Numbers and algebra

The set $\mathbb{R}$ of real numbers has an algebraic structure under the usual addition and multiplication rules for real numbers. $\mathbb{R}$ contains the set of natural numbers
$$\mathbb{N}={1,2, \ldots, n, \ldots} .$$
The computational rules of $\mathbb{R}$ may also be applied to $\mathbb{N}$ because sums and products of two natural numbers yield natural numbers. ${ }^7$ Similar algebraic rules can be defined on other sets. We give two examples below.

Remark 1.6. There is the philosophical issue whether “0” is a natural number, which corresponds to the question whether an entity can be a “set” when it is cmpty, i.c., contains no clement. 8 For clarification, we therefore employ the notation
$$\mathbb{N}_0=\mathbb{N} \cup{0}$$
for the set of natural numbers including 0 .
Complex numbers. There is no real number $r \in \mathbb{R}$ with the property $r^2=-1$. To remedy this deficiency, one may introduce a new “number” i and do computations with it like it were a real number with the property
$$i^2=-1$$

In doing so, one arrives at more general numbers of the form $z=$ $a+\mathrm{i} b$, with $a$ and $b$ being real numbers. The set
$$\mathbb{C}={a+\mathrm{i} b \mid a, b \in \mathbb{R}}$$
is the set of complex numbers. The special number
$$\mathrm{i}=0+\mathrm{i} \cdot 1$$
is the so-called imaginary unit. Using the algebraic rules of $\mathbb{R}$ and always keeping $\mathrm{i}^2=-1$ in mind, one can perform the usual computations with additions, subtractions, multiplications and divisions in $\mathbb{C}$.

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|Algebra of functions and matrices

$$\mathbb{R}^S=f: S \rightarrow \mathbb{R}$$

## 经济代写|博弈论代写Game Theory代考|Numbers and algebra

$$\mathbb{N}=1,2, \ldots, n, \ldots .$$

$$\mathbb{N}_0=\mathbb{N} \cup 0$$

$$i^2=-1$$

$$\mathbb{C}=a+\mathrm{i} b \mid a, b \in \mathbb{R}$$

$$\mathrm{i}=0+\mathrm{i} \cdot 1$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 经济代写|博弈论代写Game Theory代考|ECON40010

statistics-lab™ 为您的留学生涯保驾护航 在代写博弈论Game Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写博弈论Game Theory代写方面经验极为丰富，各种代写博弈论Game Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|博弈论代写Game Theory代考|Mathematical modelling

Mathematics is the powerful human instrument to analyze and to structure observations and to possibly discover natural “laws”. These laws are logical principles that allow us not only to understand observed phenomena (i.e., the so-called real world) but also to compute possible evolutions of current situations, and thus to attempt a “look into the future”.

Why is that so? An answer to this question is difficult if not impossible. There is a wide-spread belief that mathematics is the language of the universe. ${ }^1$ So everything can supposedly be captured by mathematics and all mathematical deductions reveal facts about the real world. I do not know whether this is true. Even if it were, one would have to be careful with real-world interpretations of mathematics, nonetheless. A simple example may illustrate the difficulty:
While apples on a tree are counted in terms of natural numbers, it would certainly be erroneous to conclude: for every natural number $n$, there exists a tree with $n$ apples. In other words, when we use the set of nonnegative integers to describe the number of apples on a tree, our mathematical model will comprise mathematical objects that have no real counterparts.

Theoretically, one could try to get out of the apple dilemma by restricting the mathematical model to those numbers $n$ that are realized by apple trees. But such a restricted model would be of no practical use as neither the set of such apple numbers $n$ nor its specific algebraic structure is explicitly known. Indeed, while the sum $m+n$ of two natural numbers $m$ and $n$ is a natural number, it is not clear whether the existence of two apple trees with $m$ resp. $n$ apples guarantees the existence of an apple tree with $m+n$ apples.

In general, a mathematical model of a real-world situation is, alas, not necessarily guaranteed to be absolutely comprehensive. Mathematical conclusions are possibly only theoretical and may suggest objects and situations which do not exist in reality. One always has to double-check real-world interpretations of mathematical deductions and ask whether an interpretation is “reasonable” in the sense that it is commensurate with one’s own personal experience.

In the analysis of a game-theoretic situation, for example, one may want to take the psychology of individual players into account. A mathematical model of psychological behavior, however, is typically based on assumptions whose accuracy is unclear. Consequently, mathematically established results within such models must be interpreted with care, of course.

## 经济代写|博弈论代写Game Theory代考|Functions and data representation

A function $f: S \rightarrow W$ assigns elements $f(s)$ of a set $W$ as values to the elements $s$ of a set $S$. One way of looking at a function is to imagine a measuring device ” $f$ ” which produces the result $f(s)$ upon the input $s$ :
$$s \in S \longrightarrow f \rightarrow f(s) \in W .$$
We denote the collection of all $W$-valued functions with domain $S$ as
$$W^S={f: S \rightarrow W}$$ and think of an element $f \in W^S$ also as a parameter vector whose coordinates $f_s$ are indexed by the elements $s \in S$ and have values $f_s=f(s) \in W$.

There is a dual way of looking at this situation where the roles of the function $f$ and the variable $s$ are reversed. The dual viewpoint sees $s$ as a probe which produces the value $f(s)$ when exposed to $f$ :
If $S$ is small, the function $f$ can be presented by a table which displays the total effect of $f$ on $S$ :

The dual viewpoint would fix an element $s \in S$ and evaluate the effect of the measuring devices $f_1, \ldots, f_k$, for example, and thus represent an individual element $s \in S$ by a $k$-dimensional data table:
$s \longleftrightarrow$\begin{tabular}{|c|c|c|c|c|}
\hline$f_1$ & $f_2$ & $f_3$ & $\ldots$ & $f_k$ \
\hline$f_1(s)$ & $f_2(s)$ & $f_3(s)$ & $\ldots$ & $f_k(s)$ \
\hline
\end{tabular}
The dual viewpoint is typically present when one tries to describe the state $\sigma$ of an economic, social or physical system $\mathfrak{S}$ via the data values $f_1(\sigma), f_2(\sigma), \ldots, f_k(\sigma)$ of statistical measurements $f_1, \ldots, f_k$ with respect to $k$ system characteristics:
$$\sigma{} \quad\left(f_1(\sigma), f_2(\sigma), \ldots, f_k(\sigma)\right) .$$
The two viewpoints are logically equivalent. Indeed, the dual perspective sees the element $s \in S$ just like a function $\hat{s}: W^S \rightarrow W$ with values
$$\hat{s}(f)=f(s) .$$

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|Functions and data representation

$$s \in S \longrightarrow f \rightarrow f(s) \in W .$$

$$W^S=f: S \rightarrow W$$

$s \longleftrightarrow$

$$\sigma \quad\left(f_1(\sigma), f_2(\sigma), \ldots, f_k(\sigma)\right) .$$

$$\hat{s}(f)=f(s)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 经济代写|博弈论代写Game Theory代考|ECOS3012

statistics-lab™ 为您的留学生涯保驾护航 在代写博弈论Game Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写博弈论Game Theory代写方面经验极为丰富，各种代写博弈论Game Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|博弈论代写Game Theory代考|Cooperative TU-games

A transferable utility relative to a set $N$ of players is a quantity $v$ whose value $v(S)$ depends on the coalition $S$ of active players and hence is a potential
$$v: \mathcal{N} \rightarrow \mathbb{R} .$$
The potential $v$ implies the utility measure $\partial v: \mathcal{N} \times \mathcal{N} \rightarrow \mathbb{R}$ with the values
$$\partial v(S, T)=v(T)-v(S) .$$
We denote resulting potential game by $\Gamma=(N, v)$ and refer to it as a cooperative TU-game with characteristic function $v$.

Ex. 8.1. Assume that the players $i \in N$ evaluate their utility relative to a coalition $S \subseteq N$ by a real parameters $u_i(S)$, which means that each player $i \in N$ has an individual utility function
$$u_i: \mathcal{N} \rightarrow \mathbb{R} .$$
The aggregated utility $u=\sum_{i \in N} u_i$ is then a transferable utility and defines the TU-game $\Gamma=(N, u)$. For each $S \subseteq N$ and player $i \in N$, one finds:
\begin{aligned} \partial u_i(S, S \cup{i}) &=u_i(S \cup{i})-u_i(S) \ &=u(S \cup{i})-u(S)=\partial u(S, S \cup{i}) . \end{aligned}

## 经济代写|博弈论代写Game Theory代考|Vector spaces of TU-games

Identifying a TU-game $(N, v)$ with its characteristic function $v$, we think of the function space
$$\mathbb{R}^{\mathcal{N}}={v: \mathcal{N} \rightarrow \mathbb{R}} \quad \text { with } \quad \mathcal{N}={S \subseteq N}$$
as the vector space of all (not necessarily zero-normalized) TU-games on the set $N . \mathbb{R}^{\mathcal{N}}$ is isomorphic with coordinate space $\mathbb{R}^{2^n}$ and has dimension
$$\operatorname{dim} \mathbb{R}^{\mathcal{N}}=|\mathcal{N}|=2^n=\operatorname{dim} \mathbb{R}^{2^n} .$$
The $2^n$ unit vectors of $\mathbb{R}^{\mathcal{N}}$ correspond to the so-called DiraC functions $\delta_S \in \mathbb{R}^{\mathcal{N}}$ with the values
$$\delta_S(T)= \begin{cases}1 & \text { if } T=S \ 0 & \text { if } T \neq S .\end{cases}$$
The set $\left{\delta_S \mid S \in \mathcal{N}\right}$ is a basis of $\mathbb{R}^{\mathcal{N}}$. Any $v \in \mathbb{R}^{\mathcal{N}}$ has the representation
$$v=\sum_{S \in \mathcal{N}} v(S) \delta_S$$

## 经济代写|博弈论代写Game Theory代考|Cooperative TU-games

$$v: \mathcal{N} \rightarrow \mathbb{R} .$$

$$\partial v(S, T)=v(T)-v(S) .$$

$$u_i: \mathcal{N} \rightarrow \mathbb{R} .$$

$$\partial u_i(S, S \cup i)=u_i(S \cup i)-u_i(S) \quad=u(S \cup i)-u(S)=\partial u(S, S \cup i)$$

## 经济代写|博弈论代写Game Theory代考|Vector spaces of TU-games

$$\mathbb{R}^{\mathcal{N}}=v: \mathcal{N} \rightarrow \mathbb{R} \quad \text { with } \quad \mathcal{N}=S \subseteq N$$

$$\operatorname{dim} \mathbb{R}^{\mathcal{N}}=|\mathcal{N}|=2^n=\operatorname{dim} \mathbb{R}^{2^n} .$$

$$\delta_S(T)={1 \quad \text { if } T=S 0 \quad \text { if } T \neq S .$$
$$v=\sum_{S \in \mathcal{N}} v(S) \delta_S$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 经济代写|博弈论代写Game Theory代考|ECON6025

statistics-lab™ 为您的留学生涯保驾护航 在代写博弈论Game Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写博弈论Game Theory代写方面经验极为丰富，各种代写博弈论Game Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|博弈论代写Game Theory代考|Temperature of matrix games

Let $\Gamma=\Gamma\left(u_i \mid i \in N\right)$ be an $n$-person game with player set $N=$ ${1, \ldots, n}$ where each player $i \in N$ has a finite set $X_i$ of strategic resources and a utility function
$$u_i: \mathfrak{X} \rightarrow \mathbb{R} \quad \text { (with } \mathfrak{X}=X_1 \times X_2 \times \ldots \times X_n \text { ) }$$
In the model of randomized matrix games, it is assumed that the players $i$ choose probability distributions $\pi^{(i)}$ on their strategy sets $X_i$ independently from each other and then select elements $x_i \in X_i$ according to those distributions.

Let us drop the stochastic independence assumption and consider the more general model where the joint strategy
$$\mathbf{x}=\left(x_1, x_2, \ldots, x_n\right) \in \mathfrak{X}$$
would be chosen by the player set $N$ with a certain probability $\pi_{\mathbf{x}}$. The aggregated total utility value is then expected to be
$$\mu=\sum_{\mathbf{x} \in \mathfrak{X}} \sum_{i \in N} u_i(\mathbf{x}) \pi_{\mathbf{x}} .$$
The players’ total utility
$$u(\mathbf{x})=\sum_{i \in N} u_i(\mathbf{x})$$ is a potential on $\mathfrak{X}$. So one may consider the (BoltZmanN) temperature relative to $u$. In the case
$$\mu=\frac{1}{Z_T} \sum_{\mathbf{x} \in \mathfrak{X}} e^{u(\mathbf{x}) / T} \quad\left(\text { with } Z_T=Z(1 / T)\right)$$
we say that $\Gamma$ is is played at temperature $T$. If $|T| \approx \infty$ (i.e., $|T|$ is very large), we expect about the average value of the total utility:
$$\mu \approx \frac{1}{|\mathfrak{X}|} \sum_{\mathbf{x} \in \mathfrak{X}} u(\mathbf{x}) .$$
If $T>0$ is very small (i.e., $T \approx 0$ ), then we may expect about the maximal total utility:
$$\mu \approx \max {\mathbf{x} \in \mathfrak{X}} u(\mathbf{x}) .$$ Similarly, if $T \approx 0$ and $T<0$ holds, about the minimal total utility value is to be expected: $$\mu \approx \min {\mathbf{x} \in \mathfrak{X}} u(\mathbf{x}) .$$

## 经济代写|博弈论代写Game Theory代考|Cooperative Games

While the agents in the $n$-person games of the previous chapters typically have individual utility objectives and thus possibly opposing strategic goals, the model of a cooperative game refers to a finite set $N$ of $n=|N|$ players that may or may not be active towards a common goal. A subset $S \subseteq N$ of potentially active players is traditionally called a coalition. Mathematically, there are several ways of looking at the system of coalitions:

From a set-theoretic point of view, one has the system of the $2^n$ coalitions
$$\mathcal{N}={S \mid S \subseteq N} .$$

On the other hand, one may represent a subset $S \in \mathcal{N}$ by its incidence vector $x^{(S)} \in \mathbb{R}^N$ with the coordinates
$$x_i^{(S)}= \begin{cases}1 & \text { if } i \in S \ 0 & \text { if } i \notin S .\end{cases}$$
The incidence vector $x^{(S)}$ suggests the interpretation of an “activity vector”:
$i \in N$ is active if $x_i^{(S)}=1$.
The coalition $S$ would thus be the collection of active players.
A further interpretation imagines every player $i \in N$ to have a binary strategy set $X_i={0,1}$ from which to choose one element. An incidence vector
$$x=\left(x_1, \ldots, x_n\right) \in X_1 \times \cdots \times X_n={0,1}^N \subseteq \mathbb{R}^N$$
represents the joint strategy decision of the $n$ players and we have the correspondence
$$\mathcal{N} \longleftrightarrow{0,1}^N=2^N$$
By a cooperative game we will just understand a $n$-person game $\Gamma$ with player set $N$ and state set
$$\mathfrak{X}=\mathcal{N} \text { or } \mathfrak{X}=2^N,$$
depending on a set-theoretic or on a vector space point of view. A general cooperative game $\Gamma=\left(u_i \mid i \in N\right)$ with individual utility functions $u_i: \mathcal{N} \rightarrow \mathbb{R}$ is therefore a matrix game where each player has the choice between two alternative actions.

## 经济代写|博弈论代写Game Theory代考|Temperature of matrix games

$u_i: \mathfrak{X} \rightarrow \mathbb{R} \quad$ (with $\mathfrak{X}=X_1 \times X_2 \times \ldots \times X_n$ )

$$\mathbf{x}=\left(x_1, x_2, \ldots, x_n\right) \in \mathfrak{X}$$

$$u(\mathbf{x})=\sum_{i \in N} u_i(\mathbf{x})$$

$$\mu=\frac{1}{Z_T} \sum_{\mathbf{x} \in \mathfrak{X}} e^{u(\mathbf{x}) / T} \quad\left(\text { with } Z_T=Z(1 / T)\right)$$

$$\mu \approx \frac{1}{|\mathfrak{X}|} \sum_{\mathbf{x} \in X} u(\mathbf{x}) .$$

$$\mu \approx \max \mathbf{x} \in \mathfrak{X} u(\mathbf{x}) .$$

$$\mu \approx \min \mathbf{x} \in \mathfrak{X} u(\mathbf{x})$$

## 经济代写|博弈论代写Game Theory代考|Cooperative Games

$$\mathcal{N}=S \mid S \subseteq N .$$

$$x_i^{(S)}={1 \quad \text { if } i \in S 0 \quad \text { if } i \notin S .$$

$i \in N$ 是活跃的，如果 $x_i^{(S)}=1$.

$$x=\left(x_1, \ldots, x_n\right) \in X_1 \times \cdots \times X_n=0,1^N \subseteq \mathbb{R}^N$$

$$\mathcal{N} \longleftrightarrow 0,1^N=2^N$$

$$\mathfrak{X}=\mathcal{N} \text { or } \mathfrak{X}=2^N,$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 经济代写|博弈论代写Game Theory代考|ECON2112

statistics-lab™ 为您的留学生涯保驾护航 在代写博弈论Game Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写博弈论Game Theory代写方面经验极为丰富，各种代写博弈论Game Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|博弈论代写Game Theory代考|BOLTZMANN temperature

From Lemma 7.1, it is clear that one could characterize the expected value $\mu$ of a non-constant potential $v$ equally well by specifying the parameter $t \in \mathbb{R} \cup{-\infty,+\infty}$ of the BoltZmanN distribution $\beta(t)$ with expectation
$$\mu(t)=\mu .$$
In analogy with the BoltZMANN model in statistical thermodynamics for the temperature, we call the related parameter
$$T=1 / t .$$
the temperature of the system $\mathfrak{S}$ relative to a potential with the expected value $\mu(1 / T)$. Adjusting the notation accordingly to
$$\beta^{(T)}=\beta(1 / T) \quad \text { and } \quad \mu^{(T)}=\mu(1 / T) .$$
the BOLTZMANN distribution $\beta^{(T)}$ has the coefficients
$$\beta_\sigma^{(T)}=\frac{e^{v_\sigma / T}}{\sum_{\tau \in \mathbb{S}} e^{v_\tau / T}} \quad(\sigma \in \mathcal{S}) .$$
As the system “freezes” to the temperature $T=0$, one obtains the extreme values of the potential $v$ as the expectations in the limit, depending on whether the limit 0 is approached from the positive or the negative side:
\begin{aligned} &\lim {T \rightarrow 0^{+}} \mu^{(T)}=\max {\sigma \in \mathfrak{S}} v_\sigma \ &\lim {T \rightarrow 0^{-}} \mu^{(T)}=\min {\sigma \in \mathbb{S}} v_\sigma . \end{aligned}
In contrast, all states of $\mathfrak{S}$ are equally likely at when the temperature $T$ is infinite.

## 经济代写|博弈论代写Game Theory代考|The METROPOLIS process

METROPOLIS et al. [29] have pointed out that BoltZMANN distributions may result dynamically as limiting distributions of quite general stochastic processes.

To formulate the process, we associate with each $\sigma \in \mathfrak{S}$ a neighborhood $\mathcal{F}(\sigma) \subseteq \mathfrak{S}$ that renders $\mathfrak{S}$ connected in the following sense:

• For any $\sigma, \tau \in \mathfrak{S}$, there are $\sigma_1, \ldots, \sigma_k \in \mathfrak{S}$ such that $\sigma_{\ell+1} \in \mathcal{F}\left(\sigma_{\ell}\right)$ holds for all $\ell=0, \ldots k$, where $\sigma_0=\sigma$ and $\sigma_{k+1}=\tau$.
• It follows that the METropolis state transitions define a MARKOV chain ${ }^4$ on $\mathfrak{S}$ with transition probabilities
• $$• p_{\sigma \tau}=\frac{q_{\sigma \tau}}{|\mathcal{F}(\sigma)|} . •$$
• The Metropolis process converges as a Markov chain to a limiting distribution on $\mathfrak{S}$ under quite general conditions. A sufficient condition is, for example:
• Proof. By Ex. A.12 of Section 7 of the Appendix, it suffices to check that the condition
• $$• \beta_\sigma(t) p_{\sigma \tau}=\beta_\tau(t) p_{\tau \sigma} . •$$
• is satisfied by any $\sigma, \tau \in \mathfrak{S}$. So assume $v_\tau<v_\sigma$, for example. Then
• $$• \beta_\sigma(t) p_{\sigma \tau}=\frac{e^{v_\sigma t} e^{\left(v_\tau-u_\sigma\right) t}}{Z(t) f}=\frac{e^{v_\tau t}}{Z(t) f}=\beta_\tau(t) p_{\tau \sigma} . •$$
• Simulated annealing. The MEtropolis process suggests a simple intuitive method for maximizing a function $v: X \rightarrow \mathbb{R}$ over a finite set $X$ :
• $$• \max _{x \in X} v_x . •$$

## 经济代写|博弈论代写Game Theory代考|BOLTZMANN temperature

$$\mu(t)=\mu .$$

$$T=1 / t .$$

$$\beta^{(T)}=\beta(1 / T) \quad \text { and } \quad \mu^{(T)}=\mu(1 / T) .$$

$$\beta_\sigma^{(T)}=\frac{e^{v_\sigma / T}}{\sum_{\tau \in \mathbb{S}} e^{v_\tau / T}} \quad(\sigma \in \mathcal{S}) .$$

$$\lim T \rightarrow 0^{+} \mu^{(T)}=\max \sigma \in \mathcal{S} v_\sigma \quad \lim T \rightarrow 0^{-} \mu^{(T)}=\min \sigma \in \mathbb{S} v_\sigma .$$

## 经济代写|博弈论代写Game Theory代考|The METROPOLIS process

• 对于任何 $\sigma, \tau \in \mathcal{S}$ ，有 $\sigma_1, \ldots, \sigma_k \in \mathfrak{S}$ 这样 $\sigma_{\ell+1} \in \mathcal{F}\left(\sigma_{\ell}\right)$ 适用于所有人 $\ell=0, \ldots k$ ，在哪里 $\sigma_0=\sigma$ 和 $\sigma_{k+1}=\tau$.
• 由此可见, MEtropolis 状态转换定义了一个 MARKOV 链 ${ }^4$ 上؟有转移概率
• $\$ \$$• \ \$$
• Metropolis 过程作为马尔可夫链收敛到有限分布
• 证明。由前。附录第 7 节的 A.12,只需检查条件
• $\$ \$$• \backslash beta_Isigma(t) p_{-}{\backslash sigma \backslash tau }=\backslash b^{-}beta_Itau(t) ^2{\backslash \operatorname{tau} \backslash sigma 。 • \ \$$
• 满足任何 $\sigma, \tau \in \subseteq$. 所以假设 $v_\tau<v_\sigma$ ， 例如。然后
• $\$ \$$• \backslash beta_Isigma(t) p_{-}{\backslash sigma \backslash t a u}=\backslash f r a c\left{e^{\wedge}\left{v_{-}\right.\right.Isigma \left.t\right} e^{\wedge}{\backslash left(v_Itau-u_Isigma \backslash right \left.\left.) t\right}\right}{Z(t) f}=\backslash f r a c\left{e^{\wedge}\left{v_{-}\right.\right.Itau \left.\left.t\right}\right}{Z(t) f}=\backslash beta_Itau(t) p_{-}{Itau \backslash sigma } 。 • \ \$$
• 模拟退火。MEtropolis 过程提出了一种简单直观的方法来最大化函数 $v: X \rightarrow \mathbb{R}$ 在有限集上 $X:$
• $\$ \$$• \backslash \max {x \backslash \operatorname{in} X} v_{-} x 。 • \ \$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 经济代写|博弈论代写Game Theory代考|ECON2070

statistics-lab™ 为您的留学生涯保驾护航 在代写博弈论Game Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写博弈论Game Theory代写方面经验极为丰富，各种代写博弈论Game Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|博弈论代写Game Theory代考|Sums of Nim Heaps

In this section, we derive how to compute the Nim value for a general Nim position, which is a sum of different Nim heaps. This will be the Nim sum that we have defined using the binary representation, now cast in the language of game sums and equivalent games, and without assuming the binary representation.

For example, we know that $* 1+* 2+* 3 \equiv 0$, so by Lemma $1.12, * 1+* 2$ is equivalent to $* 3$. In general, however, the sizes of the Nim heaps cannot simply be added to obtain the equivalent Nim heap, because $* 2+* 3$ is also equivalent to $* 1$, and $* 1+* 3$ is equivalent to $* 2$.

If $* k \equiv * n+* m$, then we call $k$ the Nim sum of $n$ and $m$, written $k=n \oplus m$. The following theorem states that the Nim sum of distinct powers of two is their arithmetic sum. For example, $1=2^0$ and $2=2^1$, so $1 \oplus 2=1+2=3$.

Theorem 1.15. Let $n \geq 1$, and $n=2^a+2^b+2^c+\cdots$, where $a>b>c>\cdots \geq 0$. Then
$$• n \equiv \left(2^a\right)+\left(2^b\right)+*\left(2^c\right)+\cdots .$$
We first discuss the implications of this theorem, and then prove it. The expression $n=2^a+2^b+2^c+\cdots$ is an arithmetic sum of distinct powers of two. Any $n$ is uniquely given as such a sum. It amounts to the binary representation of $n$, which, if $n<2^{a+1}$, gives $n$ as the sum of all powers $2^a, 2^{a-1}, 2^{a-2}, \ldots, 2^0$ where each power of two is multiplied with 0 or 1 , the binary digit for the respective position. For example,
$$9=8+1=1 \cdot 2^3+0 \cdot 2^?+0 \cdot 2^1+1 \cdot 2^n,$$
so that 9 in decimal is written as 1001 in binary. Theorem $1.15$ uses only the distinct powers of two $2^a, 2^b, 2^c, \ldots$ that correspond to the digits 1 in the binary representation of $n$.

## 经济代写|博弈论代写Game Theory代考|Finding Nim Values

In this section, we analyze some impartial games using the mex rule in Theorem $1.14$

A game similar to the Rook-move game is the Queen-move game shown in Figure $1.3$ where the rook is replaced by a Chess queen, which may move horizontally, vertically, and diagonally (left or up). The squares on the main diagonal are therefore no longer losing positions. This game can also be played with two heaps of tokens where in one move, the player may either remove tokens from one heap as in Nim, or reduce both heaps by the same number of tokens (so this is no longer a sum of two Nim heaps!). In order to illustrate that we are not just interested in the winning and losing squares, we add to this game a Nim heap of size 4.

Figure $1.4$ shows the equivalent Nim heaps for the positions of the Queen-move game, determined by the mex rule. The square in row 3 and column 4 occupied by the queen in Figure $1.3$ has entry $* 2$. So a winning move is to remove two tokens from the Nim heap to turn it into the heap $* 2$, creating the losing position $* 2+* 2$. Because 2 is the mex of the Nim values of the options of the queen, these may include (as in Poker Nim) higher Nim values. Indeed, the queen can reach two positions equivalent to $* 4$, in row 3 column 1 , and row 0 column 4 . If the queen moves there this creates the game sum $* 4+* 4$ which is losing, so these are two further winning moves in Figure $1.3$.

The impartial game Kayles is played as follows: Given a row of $n$ bowling pins (numbered 1,2,…,n), a move knocks out one or two consecutive pins, as in this example of 5 pins where pins 2 and 3 are knocked out:

If this game is called $K_n$, then knocking out a single pin $p$ creates the game sum $K_{p-1}+K_{n-p}$, and knocking out pins $p$ and $p+1$ (in the picture, $p=2$ and $n=5$ ) creates the game sum $K_{p-1}+K_{n-p-1}$ (here $K_1+K_2$ ). The options of $K_n$ are these game sums for all possible $p$, where $p$ only needs to be considered up to the middle pin due to the symmetry of the row of pins. Note that in this game, options happen to be sums of games. For the mex rule, only their Nim value is important.

## 经济代写|博弈论代写Game Theory代考|Finding Nim Values

Kayles 的公平博亦如下：给定一排 $n$ 保龄球瓶（编号 $1,2, \ldots, \mathrm{n})$ ，一个动作会击倒一个或两个连续的保龄球瓶， 如本例中的 5 个保龄球瓶，其中 2 号和 3 号球瓶被击倒:

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。