We consider a weighted network design game, where selfish players chooses paths in a network to minimize their cost. The cost function of each edge in the network is affine linear, namely c_e(W_e) = a_eW_e + b_e, where a_e, b_e > 0 are only related to the edge e and We is the total weight of the players that choose a path containing edge e. We first show the existence of \alpha-approximate Nash equilibrium and prove the upper bound of \alpha is O(log2(W)), where W is the sum of the weight of all players. Furthermore, considering that compute the \alpha-approximate Nash equilibrium is PLS-complete, we assume that {ae, be}_{e\in E} are \phi-smooth random variables on [0, 1]. In this case, we show that \epsilon-better response dynamics can compute the {\alpha}-approximate Nash Equilibrium in polynomial time by proving the expected number of iterations is polynomial in 1/\epsilon, \phi, the number of players and the number of edges in the network.

翻译：考虑一个加权网络设计博弈，其中自私的玩家选择网络中的路径以最小化其成本。网络中每条边的成本函数是仿射线性的，即 c_e(W_e) = a_eW_e + b_e，其中 a_e, b_e > 0 仅与边 e 相关，W_e 是选择包含边 e 的路径的玩家总权重。我们首先证明 α-近似纳什均衡的存在性，并给出 α 的上界为 O(log2(W))，其中 W 是所有玩家权重之和。进一步地，考虑到计算 α-近似纳什均衡是 PLS-完全的，我们假设 {a_e, b_e}_{e∈E} 是 [0, 1] 上的 φ-光滑随机变量。在此情况下，我们证明 ε-更优响应动力学能够在多项式时间内计算出 α-近似纳什均衡，通过证明期望迭代次数关于 1/ε、φ、玩家数量和网络中边数量均为多项式。