### 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|Mathematical and Scholarly Level

General mathematical prerequisites are the basic notions of linear algebra (vectors and matrices), probability theory (independence, conditional probability), and analysis (continuity, closed sets). Rather than putting them in a rarely read appendix, important concepts are recalled where needed in text boxes labeled as Background material.

For each chapter, the necessary mathematical background and the required previous chapters are listed in a first section on prerequisites and learning objectives.
The mathematics of game theory is not technically difficult, but very conceptual, and requires therefore a certain mathematical maturity. For example, combinatorial games have a recursive structure, for which a generalization of the mathematical induction known for natural numbers is appropriate, called “top-down induction” and explained in Section 1.3.

Game-theoretic concepts have precise and often unfamiliar mathematical definitions. In this book, each main concept is typically explained by means of a detailed introductory example. We use the format of definitions, theorems and proofs in order to be precise and to keep some technical details hidden in proofs. The proofs are detailed and complete and can be studied line by line. The main idea of each proof is conveyed by introductory examples, and geometric arguments are supported by pictures wherever possible.

Great attention is given to details that help avoid unnecessary confusions. For example, a random event with two real values $x$ and $y$ as outcomes will be described with a probability $p$ assigned to the second outcome $y$, so that the interval $[0,1]$ for $p$ corresponds naturally to the interval $[x, y]$ of the expectation $(1-p) x+p y$ of that event.

This book emphasizes accessibility, not generality. I worked hard (and enjoyed) on presenting the most direct and overhead-free proof of each result. In studying these proofs, students may encounter for the first time important ideas from topology, convex analysis, and linear programming, and thus may become interested in the more general mathematical theory of these subjects.

## 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 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.

## 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。