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# 经济代写|博弈论代考Game theory代写|ECON7062 Definition of Congestion Games

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## 经济代写|博弈论代考Game theory代写|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 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|博弈论代考Game theory代写|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
$$\Phi\left(f^{Q_i}\right)-\Phi(f)=\sum_{e \in Q_i} c_e\left(f_e^{Q_i}\right)-\sum_{e \in P_i} c_e\left(f_e\right) .$$

# 博弈论代写

## 经济代写|博亦论代考Game theory代写|Definition of Congestion Games

$$\sum_{e \in P_i} c_e\left(f_e\right) .$$

## 经济代写|博亦论代考Game theory代写|Existence of Equilibrium in a Congestion Game

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

$$\Phi\left(f^{Q_i}\right)-\Phi(f)=\sum_{e \in Q_i} c_e\left(f_e^{Q_i}\right)-\sum_{e \in P_i} c_e\left(f_e\right)$$

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