### 经济代写|博弈论代写Game Theory代考|ECON 3503

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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|Definition of Congestion Games

In this section, we give a general definition of congestion games and of the concept of an equilibrium. A congestion network has the following components:

• A finite set of nodes.
• A finite collection $E$ of edges. Each edge $e$ is an ordered pair, written as $u v$, from some node $u$ to some node $v$, which is graphically drawn as an arrow from $u$ to $v$. Parallel edges (that is, with the same pair $u v$ ) are allowed (hence the edges form a “collection” $E$ rather than a set, which would not allow for such repetitions), as in Figure 2.1.
• Each edge $e$ in $E$ has a cost function $c_{e}$ that gives a value $c_{e}(x)$ when there are $x$ users on edge $e$, which describes the same cost to each user for using $e$. Each cost function is weakly increasing, that is, $x \leq y$ implies $c_{e}(x) \leq c_{e}(y)$.
• A number $N$ of $u$ sers of the network. Each user $i=1, \ldots, N$ has an origin $o_{i}$ and destination $d_{i}$, which are two nodes in the network, which may or may not be the same for all users (if they are the same, they are usually called $o$ and $d$ as in the above examples).

The underlying structure of nodes and edges is called a directed graph or digraph (where edges are sometimes called “arcs”). In such a digraph, a path $P$ from $u$ to $v$ is a sequence of distinct nodes $u_{0}, u_{1}, \ldots, u_{m}$ for $m \geq 0$ where $u_{k} u_{k+1}$ is an edge for $0 \leq k<m$, and $u=u_{0}$ and $v=u_{m}$. For any such edge $e=u_{k} u_{k+1}$ for $0 \leq k<m$ we write $e \in P$. Note that a node may appear at most once in a path. Every user $i$ chooses a path (which we have earlier also called a “route”) from her origin $o_{i}$ to her destination $d_{i}$.

• A strategy of user $i$ is a path $P_{i}$ from $o_{i}$ to $d_{i}$.
• Given a strategy $P_{i}$ for each user $i$, the load on or flow through an edge $e$ is defined as $f_{e}=\left|\left{i \mid e \in P_{i}\right}\right|$, which is the number of chosen paths that contain $e$, that is, the number of users on $e$. The cost to user $i$ for her strategy $P_{i}$, given that the other users have chosen their strategies, is then
$$\sum_{e \in P_{f}} c_{e}\left(f_{e}\right) .$$

## 经济代写|博弈论代写Game Theory代考|Atomic and Splittable Flow, Price of Anarchy

The presented model of congestion games with finitely many users on a network is only one of many ways of modeling congestion by selfish routing.

First, the representation of costs in (2.1) is a simplification in the sense that it assumes that user $i$ creates a “flow” of 1 along its chosen path, and thus contributes to the congestion on every edge on that path. This model does not take any temporal aspects into account, like the fact that the user can be only on one edge at a time but not everywhere on the path at once. Nevertheless, this is not completely unrealistic because everywhere along her chosen route, the user will create congestion at some point in time together with the other users. The flow model is very appropriate for internet routing, for example, with a continuous flow of data packets that consume bandwidth along the taken route.

The specific model that we consider here is called atomic (or non-splittable) flow where single users decide on their paths from their origin to their destination.
In contrast, splittable flow means that a flow unit is just some “mass” of users of which any parts can be routed along different paths. For example, in Figure $2.1$ the flow of 2 from $o$ to $d$ could be split into a flow of $0.01$ on the top edge and a flow of $1.99$ on the bottom edge. This would still not be an equilibrium because any one user from the $0.01$ fraction of users on the top edge could improve her cost by moving to the bottom edge (each user is negligibly small). Splittable flows are in some sense simpler because there will be no minor variation of equilibria, such as the distinct equilibria with a flow of either $y=100$ or $y=99$ on the bottom edge in Figure 2.2; the only equilibrium flow would be $y=100$. The equilibrium condition is then simply that there is no alternative path with currently smaller cost, without taking into account that there is an increased cost by switching to another path, as in the term $c_{e}\left(f_{e}+1\right)$ in (2.2) where the increase of $f_{e}$ by 1 results from that switch. In the splittable flow model, users have negligible size, so there is no such increase. It can be shown that the cost of an equilibrium flow in a splittable flow network is unique (however, there may be several equilibria with that cost, for example with an undetermined flow across two parallel edges with the same constant cost). For splittable flow, the proof of Theorem $2.2$ can be amended with a potential function $\Phi$ defined with an integral $\int_{0}^{f_{e}} c_{e}(x) d x$ that replaces the sum $c_{e}(1)+c_{e}(2)+\cdots+c_{e}\left(f_{e}\right)$ in $(2.3)$; see Roughgarden (2016, section 13.2.2).

We have used material from chapter 13 of the book by Roughgarden (2016), in particular theorem $13.6$ and its notation for our proof of Theorem 2.2. Exercise $2.3$ is figure $12.3$ of that book, rotated so as to display its symmetry. The Pigou network has been described qualitatively by Pigou (1920, p. 194).

The Braess paradox is due to Braess (1968). Exercise $2.2$ discusses the actual network used by Braess. Hong, Song, and Wu (2007) do not mention the Braess paradox in the Cheonggyecheon urban regeneration project, but show some nice “before” and “after” pictures on page 235 , and add some Fengshui to this book.
The proof of Theorem $2.2$ for atomic selfish routing with the help of a potential function $\Phi$ was originally described by Rosenthal (1973). More general games

where an equilibrium is found with the help of a potential function are the potential games studied by Monderer and Shapley (1996).

The model of splittable flow is due to Wardrop (1952), and its equilibrium is often called a Wardrop equilibrium.

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

• 一组有限的节点。
• 有限集合和的边缘。每条边和是一个有序对，写成在在, 从某个节点在到某个节点在，它以图形方式绘制为从在至在. 平行边（即具有相同的对在在) 是允许的（因此边缘形成一个“集合”和而不是一组，不允许这样的重复），如图 2.1 所示。
• 每条边和在和有代价函数C和给出一个值C和(X)当有X边缘用户和，它描述了每个用户使用相同的成本和. 每个成本函数都在弱增加，即X≤是暗示C和(X)≤C和(是).
• 一个号码ñ的在网络服务器。每个用户一世=1,…,ñ有渊源○一世和目的地d一世，它们是网络中的两个节点，对于所有用户来说可能相同也可能不同（如果相同，通常称为○和d如上面的例子）。

• 用户策略一世是一条路径磷一世从○一世至d一世.
• 给定一个策略磷一世对于每个用户一世, 边缘上的负载或流过边缘和定义为f_{e}=\left|\left{i \mid e \in P_{i}\right}\right|f_{e}=\left|\left{i \mid e \in P_{i}\right}\right|，这是包含的所选路径的数量和，也就是用户数和. 用户的成本一世因为她的策略磷一世，假设其他用户已经选择了他们的策略，那么
∑和∈磷FC和(F和).

## 经济代写|博弈论代写Game Theory代考|Atomic and Splittable Flow, Price of Anarchy

Braess 悖论归因于 Braess (1968)。锻炼2.2讨论 Braess 使用的实际网络。Hong, Song, and Wu (2007) 没有提到清溪川城市更新项目中的 Braess 悖论，而是在第 235 页展示了一些漂亮的“之前”和“之后”图片，并在本书中添加了一些风水。

Monderer 和 Shapley (1996) 研究的潜在博弈是在势函数的帮助下找到均衡的。

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## MATLAB代写

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