We consider a weighted Shapley network design game, where selfish players choose paths in a network to minimize their cost. The cost function of each edge in the network is affine linear with respect to the sum of weights of the players who choose the edge. We first show the existence of \alpha-approximate pure Nash equilibrium by constructing a potential function and establish an upper bound O(log2(W)) of \alpha, where W is the sum of the weight of all players. Furthermore, we assume that the coefficients of the cost function (affine linear function) of the edge all are \phi-smooth random variables on [0, 1]. In this case, we show that \epsilon-best response dynamics can compute the (1 + \epsilon)\alpha-approximate pure Nash equilibrium (\epsilon is a positive constant close to 0) 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：我们研究加权Shapley网络设计博弈，其中自私的参与者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关于选择该边的参与者权重之和呈仿射线性关系。我们首先通过构造势函数证明α-近似纯纳什均衡的存在性，并建立α的上界O(log2(W))，其中W为所有参与者的总权重。进一步地，我们假设边成本函数（仿射线性函数）的系数均为[0,1]上的φ-光滑随机变量。在此情形下，我们证明ε-最优响应动力学能够通过多项式时间计算出(1+ε)α-近似纯纳什均衡（ε为趋近于0的正常数），通过验证期望迭代次数关于1/ε、φ、参与者数量及网络边数呈多项式关系。