### 经济代写|博弈论代写Game Theory代考|ECON90022

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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

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).

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

Congestion means that a shared resource, such as a road, becomes more costly when more people use it. In a congestion game, multiple players decide on which resource to use, with the aim to minimize their cost. This interaction defines a game because the cost depends on what the other players do.

We present congestion networks as a model of traffic where individual users choose routes from an origin to a destination. Each edge of the network is congested by creating a cost to each user that weakly increases with the number of users of that edge. The central result (Theorem 2.2) states that every congestion game has an equilibrium where no user can individually reduce her cost by changing her chosen route.

Like Chapter 1, this is another introductory chapter to an active and diverse area of game theory, where we can quickly show some important and nontrivial results. It also gives an introduction to the game-theoretic concepts of strategies and equilibrium before we develop the general theory.

Equilibrium is the result of selfish routing by the users who individually minimize their costs. This is typically not the socially optimal outcome, which could have a lower average cost. A simple example is the “Pigou network” (see Section 2.2), named after the English economist Arthur Pigou, who introduced the concept of an externality (such as congestion) to economics. Even more surprisingly, the famous Braess paradox, threated in Section 2.3, shows that adding capacity to a road network can worsen equilibrium congestion.

In Section $2.4$ we give the general definition of congestion networks. In Section $2.5$ we prove that every congestion game has an equilibrium. This is proved with a cleverly chosen potential function, which is a single function that simultaneously reflects the change in cost of every user when she changes her strategy. Its minimum over all strategy choices therefore defines an equilibrium.
In Section 2.6, we explain the wider context of the presented model. It is the discrete model of atomic (non-splittable) flow with finitely many individual users who choose their traffic routes. As illustrated by the considered examples, the resulting equilibria are often not unique when the “last” user can optimally choose between more than one edge. The limit of many users is the continuous model of splittable flow where users can be infinitesmally small fractions of a “mass” of users who want to travel from an origin to a destination. In this model the equilibrium is often unique, but its existence proof is more technical and not given here.

We also mention the Price of Anarchy that compares the worst equilibrium cost with the socially optimal cost. For further details we give references in Section 2.7.

## 经济代写|博弈论代写Game Theory代考|The Pigou Network

Getting around at rush hour takes much longer than at less busy times. Commuters take buses where people have to queue to board. There are more buses that fill the bus lanes, and cars are in the way and clog the streets because only a limited number can pass each green light. Commuters can take the bus, walk, or ride a bike, and cars can choose different routes. Congestion, which slows everything down, depends on how many people use certain forms of transport, and which route they choose. We assume that people try to choose an optimal route, but what is optimal depends on what others do. This is an interactive situation, which we call a game. We describe a mathematical model that makes the rules of this game precise.

Figure $2.1$ shows an example of a congestion network. This particular network has two nodes, an origin $o$ and a destination $d$, and two edges that both connect $o$ to $d$. Suppose there are several users of the network who all want to travel from $o$ to $d$ and can choose either edge. Each edge has a cost $c(x)$ associated with it,

which is a function that describes how costly it is for each user to use that edge when it has a flow or load of $x$ users. The top edge has constant cost $c(x)=2$, and the bottom edge has cost $c(x)=x$. The cost could for example be travel time, or incorporate additional monetary costs. The cost is the same for all users of the edge. The cost for each user of the top edge, no matter how many people use it, is always 2 , whereas the bottom edge has cost 1 for one user, 2 for two users, and so on (and zero cost for no users, but then there is no one to profit from taking that zero-cost route).

Suppose the network in Figure $2.1$ is used by two users. They can either both use the top edge, or choose a different edge each, or both use the bottom edge. If both use the top edge, both pay cost 2, but if one of them switches to the bottom edge, that user will travel more cheaply and only pay cost 1 on that edge. We say that this situation is not in equilibrium because at least one user can improve her cost by changing her action. Note that we only consider the unilateral deviation of a single user in this scenario. If both would simultaneously switch to the bottom edge, then this edge would be congested with $x=2$ and again incur cost 2 , which is no improvement to the situation that both use the top edge.

If the two users use different edges, this is an equilibrium. Namely, the user of the bottom edge currently has cost 1 but would pay cost 2 by switching to the top edge, and clearly she has no incentive to do so. And the user of the top edge, when switching from the top edge, would increase the load on the bottom edge from 1 to 2 with the resulting new cost 2 which is also no improvement for her. This proves the equilibrium property.

## 博弈论代考

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那样）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)，在处理之前给出了很好的概述

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”。

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## MATLAB代写

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