# 经济代写|博弈论代考GAME THEORY代写|ECON90022 A congestion game

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## 经济代写|博弈论代考Game theory代写|A congestion game

Let $G=(V, E)$ be a finite graph, where we think of each directed edge $e=(v, w)$ as representing a road between cities $v$ and $w$. For each edge $e=(v, w)$ let $c_e: \mathbb{N} \rightarrow \mathbb{R}^{+}$be a monotone increasing congestion function, where $c_e(k)$ is the travel time when $k$ cars are on the road from $v$ to $w$.

Consider a finite set of players $N$. Each player $i$ has to travel from some city $v_i$ to some other city $w_i$, and so has to choose a path $s_i$ that connects $v_i$ to $w_i$. We assume that the chosen paths are always simple (i.e., do not repeat edges) and so we can think of $s_i$ simply as a subset of $E$.

Consider the game $G$ in which each player $i$ ‘s set of strategies is the set of all simple paths from $v_i$ to $w_i$. Given a strategy profile $s=\left(s_1, \ldots, s_n\right)$ and an edge $e$, we denote by $n_e(s)$ the number of players who travel on $e$ :
$$n_e(s)=\left|\left{i \in N: e \in s_i\right}\right| .$$

## 经济代写|博弈论代考Game theory代写|Potential games

Let $G=\left(N,\left{S_i\right},\left{u_i\right}\right)$ be a strategic form game. We say that $G$ is a potential game if there exists a $\Phi: S \rightarrow \mathbb{R}$ with the same property as in the example above: For every $s=\left(s_{-i}, s_i\right) \in S$ and $s^{\prime}=\left(s_{-i}, s_i^{\prime}\right) \in S_i$ it holds that
$$u_i\left(s^{\prime}\right)-u_i(s)=\Phi\left(s^{\prime}\right)-\Phi(s) .$$
The proof of 9.7 applies to any finite potential game, showing that they have no infinite better response paths. Thus better response dynamics always converges to a pure Nash equilibrium for finite potential games.

# 博弈论代写

## 经济代写|博弈论代考GAME THEORY代写|A CONGESTION GAME

$\mathrm{n}_{-} \mathrm{e}(\mathrm{s})=\backslash$ left $\mid$ left ${i \backslash$ in $\mathrm{N}: \mathrm{e}$ \in s_i $\backslash$ right $} \backslash$ right $\mid$.

## 经济代写|博弈论代考GAME THEORY代 写|POTENTIAL GAMES

$$u_i\left(s^{\prime}\right)-u_i(s)=\Phi\left(s^{\prime}\right)-\Phi(s) .$$
9.7 的证明适用于任何有限潜在博恋，表明它们没有无限好的响应路径。因此，更好的响应动态总是收敛到有限潜在 博栾的纯纳什均衡。

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