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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 $u v_{\text {, }}$ 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 users 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
$$\sum_{e \in P_i} c_e\left(f_e\right) .$$

## 经济代写|博弈论代写Game Theory代考|Existence of Equilibrium in a Congestion Game

The following is the central theorem of this chapter. It is proved with the help of a potential function $\Phi$. The potential function is constructed in such a way that it defines for each edge the increase in cost created by each additional user on the edge, as explained further after the proof.

Theorem 2.2. Every congestion game (as obtained from a congestion network) has at least one equilibrium.

Proof. Suppose the $N$ strategies of the users are $P_1, \ldots, P_N$, which defines a flow $f_e$ on each edge $e \in E$, namely the number of users $i$ with $e \in P_i$. We call this the flow $f$ induced by these strategies. We now define the following function $\Phi(f)$ of this flow by
$$\Phi(f)=\sum_{e \in E}\left(c_e(1)+c_e(2)+\cdots+c_e\left(f_e\right)\right) .$$
Suppose that user $i$ changes her path $P_i$ to $Q_i$. We call the resulting new flow $f Q_i$. We will prove that
$$\Phi\left(f^{Q_i}\right)-\Phi(f)=\sum_{e \in Q_i} c_e\left(f_e^{Q_i}\right)-\sum_{e \in P_i} c_e\left(f_e\right) .$$

The right-hand side is the difference in costs to user $i$ between her strategy $Q_i$ and her strategy $P_i$, according to (2.1). Her cost for the flow $f^{Q_i}$ has been written on the right-hand side of $(2.2)$ in terms of the original flow $f$ (when she uses $P_i$ ) as
$$\sum_{e \in Q_i} c_e\left(f_e^{Q_i}\right)=\sum_{e \in Q_i \cap P_i} c_e\left(f_e\right)+\sum_{e \in Q_i \backslash P_i} c_e\left(f_e+1\right),$$
and her cost for the original flow $f$ can be expressed similarly, namely as
$$\sum_{e \in P_i} c_e\left(f_e\right)=\sum_{e \in P_i \cap Q_i} c_e\left(f_e\right)+\sum_{e \in P_i \backslash Q_i} c_e\left(f_e\right) .$$
Hence, the right-hand side of $(2.4)$ is
$$\sum_{e \in Q_i} c_e\left(f_e^{Q_i}\right)-\sum_{e \in P_i} c_e\left(f_e\right)=\sum_{e \in Q_i \backslash P_i} c_e\left(f_e+1\right)-\sum_{e \in P_i \backslash Q_i} c_e\left(f_e\right) .$$
We claim that because of (2.3) (which is why $\Phi$ is defined that way), the right-hand side of (2.7) is equal to the left-hand side $\Phi\left(f Q_i\right)-\Phi(f)$ of (2.4). Namely, by changing her path from $P_i$ to $Q_i$, user $i$ increases the flow on any new edge $e$ in $Q_i \backslash P_i$ from $f_e$ to $f_e+1$, and thus adds the term $c_e\left(f_e+1\right)$ to the sum in (2.3). Similarly, for any edge $e$ in $P_i \backslash Q_i$ which is in $P_i$ but no longer in $Q_i$, the flow $f_i^{Q_i}(e)$ is reduced from $f_e$ to $f_e-1$, so that the term $c_e\left(f_e\right)$ has to be subtracted from the sum in (2.3). This shows (2.4).

# 博弈论代考

## 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 $E$ 的边缘。每条边 $e$ 是一个有序对，写成 $u v$, 从某个节点 $u$ 到某个节点 $v$ ，它以图形方式绘制为从 $u$ 至 $v$. 平行边（即具有相同的对 $u v$ ) 是允许的（因此 边形成一个”集合“E 而不是一个集合，它不允许这样的重复)，如图 $2.1$ 所示。
• 每条边 $e$ 在 $E$ 有代价函数 $c_e$ 给出一个值 $c_e(x)$ 当有 $x$ 边缘用户 $e$ ，它描述了每个用户使用相同的成本 $e$. 每个成本函数都在弱增加，即 $x \leq y$ 暗示 $c_e(x) \leq c_e(y)$.
• 一个号码 $N$ 网络的用户。每个用户 $i=1, \ldots, N$ 有渊源 $o_i$ 和目的地 $d_i$ ，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 $o$ 和 $d$ 如上面的例子)。
节点和边的底层结构称为有向图或有向图（边有时称为“弧”) 。在这样的有向图中，路径 $P$ 从 $u$ 至 $v$ 是一系列不同的节点 $u_0, u_1, \ldots, u_m$ 为了 $m \geq 0$ 在哪里 $u_k u_{k+1}$ 是一个优势 $0 \leq k<m$ ，和 $u=u_0$ 和 $v=u_m$. 对于任何这样的边缘 $e=u_k u_{k+1}$ 为了 $0 \leq k<m$ 我们写 $e \in P$. 请注意，一个节点在路径中最多可能出现一次。 每个用户 $i$ 从她的原点选择一条路径 (我们之前也称为“路线”) $o_i$ 到她的目的地 $d_i$.
• 给定一个策略 $P_i$ 对于每个用户 $i$, 边缘上的负载或流过边缘 $e$ 定义为 $\backslash 1 e f t$ 的分隔符缺失或无法识别 用户数e. 用户的成本 $i$ 因为她的策略 $P_i$ ，假设其他用户已经选择了他们的策略，那么
，这是包含的所选路径的数量 $e$ ，也就是
$$\sum_{e \in P_i} c_e\left(f_e\right) .$$

## 经济代写|博弈论代写Game Theory代考|Existence of Equilibrium in a Congestion Game

$$\Phi(f)=\sum_{e \in E}\left(c_e(1)+c_e(2)+\cdots+c_e\left(f_e\right)\right) .$$

$$\Phi\left(f^{Q_i}\right)-\Phi(f)=\sum_{e \in Q_i} c_e\left(f_e^{Q_i}\right)-\sum_{e \in P_i} c_e\left(f_e\right) .$$
$$\sum_{e \in Q_i} c_e\left(f_e^{Q_i}\right)=\sum_{e \in Q_i \cap P_i} c_e\left(f_e\right)+\sum_{e \in Q_i \backslash P_i} c_e\left(f_e+1\right),$$

$$\sum_{e \in P_i} c_e\left(f_e\right)=\sum_{e \in P_{\vartheta} \backslash Q_i} c_e\left(f_e\right)+\sum_{e \in P_i \backslash Q_i} c_e\left(f_e\right) .$$

$$\sum_{e \in Q_i} c_e\left(f_e^{Q_i}\right)-\sum_{e \in P_i} c_e\left(f_e\right)=\sum_{e \in Q_i \backslash P_i} c_e\left(f_e+1\right)-\sum_{e \in P_i \backslash Q_i} c_e\left(f_e\right) .$$

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