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• 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|A Glimpse of Partizan Games

A combinatorial game is called partizan if the available moves in a game position may be different for the two players. These games have a rich theory of which we sketch the first concepts here, in particular the four outcome classes, and numbers as strengths of positions in certain games, pioneered by Conway (2001).

The two players are called Left and Right. Consider the game Domineering, given by a board of squares (the starting position is typically rectangular but does not have to be), where in a move Left may occupy two vertically adjacent free squares with a vertical domino, and Right two horizontally adjacent squares with a horizontal domino. (At least in lower case, the letter ” $\mathrm{l}$ ” is more vertical than the letter ” $r$ ” to remember this.) Hence, starting from a 3-row, 2-column board, we have, up to symmetry,
This game is very different from the impartial game Cram (see Exercise 1.4) where cach player may place a domino in cither orientation. $\Lambda \mathrm{s}$ (1.23) shows, in Domineering the options of Left and Right are usually different. As before, we assume the normal play convention that a player who can no longer move loses.
We always assume optimal play. In (1.23), this means Right would choose the first option and place the domino in the middle, after which Left can no longer move, whereas the second option that leaves a $2 \times 2$ board would provide Left with two further moves.

An impartial game can only be losing or winning as stated in Lemma 1.1. For partizan games, there are four possible outcome classes, which are denoted by the following calligraphic letters:
$\mathcal{L}:$ Left wins no matter who moves first.
$\mathcal{R}:$ Right wins no matter who moves first.
$\mathcal{P}$ : The first player to move loses, so the previous player wins.
$\mathcal{N}$ : The first player to move wins. (Sometimes called the “next player”, although this is as ambiguous as “next Friday”, because it is the current player who wins.)

Every game belongs to exactly one of these outcome classes. The $\mathcal{P}$-positions are what we have called losing positions, and $\mathcal{N}$-positions are what we have called winning positions. For partizan games we have to consider the further outcome classes $\mathcal{L}$ and $\mathcal{R}$. In Domineering, a single vertical strip of at least two squares, in belongs to $\mathcal{R}$. The starting $3 \times 2$ board in (1.23) belongs to $\mathcal{N}$.

An undergraduate textbook on combinatorial games is Albert, Nowakowski, and Wolfe (2007), from which we have adopted the name “top-down induction”. This book starts from many examples of games, considers in depth the game Dots and Boxes, and treats the theory of short games where (as we have assumed throughout) every position has only finitely many options and no game position is ever revisited (which in “loopy” games is allowed and could lead to a draw). An authoritative graduate textbook is Siegel (2013), which gives excellent overviews before treating each topic in depth, including newer research developments. It stresses the concept of equivalence of games that we have treated in Section 1.4.

The classic text on combinatorial game theory is Winning Ways by Berlekamp, Conway, and Guy (2001-2004). Its wit, vivid colour drawings, and the wealth of games it considers make this a very attractive original source. However, the mathematics is hard, and it helps to know first what the topic is about; for example, equivalence of games is rather implicit and written as equality.

All these books consider directly partizan games. We have chosen to focus on the simpler impartial games, apart from our short introduction to partizan games in Section 1.8. For a more detailed study of games as numbers see Albert, Nowakowski, and Wolfe (2007, Section 5.1), where Definition $5.12$ is our Definition 1.18.

The winning strategy for the game Nim based on the binary system was first described by Bouton (1901). The Queen move game is due to Wythoff (1907), described as an extension of Nim. The observation that every impartial game is equivalent to a Nim heap is independently due to Sprague (1935) and Grundy (1939). This is therefore also called the Sprague-Grundy theory of impartial games, and Nim values are also called “Sprague-Grundy values” or just “Grundy values”.
The approach in Section $1.5$ to the mex rule is inspired by Berlekamp, Conway, and Guy (2001-2004), chapter 4, which describes Poker Nim, the Rook and Queen move games, Northcott’s game (Exercise 1.8), and chapter 5 with Kayles and Lasker’s Nim (called Split-Nim in Exercise 1.12). Chomp (Exercises $1.5$ and 1.10) was described by Gale (1974); for a generalization see Gale and Neyman (1982). The digraph game in Exercise $1.9$ is another great example for understanding the mex rule, due to Fraenkel (1996).

# 博弈论代考

## 经济代写|博弈论代写Game Theory代考|A Glimpse of Partizan Games

R:不管谁先走，对的就是赢。

ñ: 第一个移动的玩家获胜。（有时称为“下一个玩家”，尽管这与“下周五”一样含糊不清，因为获胜的是当前玩家。）

Albert、Nowakowski 和 Wolfe (2007) 是一本关于组合博弈的本科教科书，我们从中采用了“自上而下的归纳法”的名称。本书从许多博弈示例开始，深入研究点与盒博弈，并处理短博弈理论，其中（正如我们自始至终假设的）每个位置只有有限多个选项，并且永远不会重新审视任何游戏位置（在“循环”游戏是允许的，可能会导致平局）。权威的研究生教科书是 Siegel (2013)，它在深入处理每个主题之前给出了很好的概述，包括新的研究进展。它强调了我们在 1.4 节中讨论过的博弈等价的概念。

Bouton（1901）首先描述了基于二进制系统的 Nim 游戏的获胜策略。皇后移动游戏是由​​于 Wythoff (1907)，被描述为 Nim 的延伸。Sprague (1935) 和 Grundy (1939) 独立地观察到每个不偏不倚的游戏都等同于 Nim 堆。因此，这也被称为 Sprague-Grundy 公平博弈理论，Nim 值也被称为“Sprague-Grundy 值”或简称为“Grundy 值”。

## 有限元方法代写

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## MATLAB代写

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