We study the convergence of best-response dynamics in Tullock contests with convex cost functions (these games always have a unique pure-strategy Nash equilibrium). We show that best-response dynamics rapidly converges to the equilibrium for homogeneous agents. For two homogeneous agents, we show convergence to an $\epsilon$-approximate equilibrium in $\Theta(\log\log(1/\epsilon))$ steps. For $n \ge 3$ agents, the dynamics is not unique because at each step $n-1 \ge 2$ agents can make non-trivial moves. We consider the model proposed by Ghosh and Goldberg (2023), where the agent making the move is randomly selected at each time step. We show convergence to an $\epsilon$-approximate equilibrium in $O(\beta \log(n/(\epsilon\delta)))$ steps with probability $1-\delta$, where $\beta$ is a parameter of the agent selection process, e.g., $\beta = n^2 \log(n)$ if agents are selected uniformly at random at each time step. We complement this result with a lower bound of $\Omega(n + \log(1/\epsilon)/\log(n))$ applicable for any agent selection process.

翻译：我们研究了凸成本函数Tullock竞赛（此类博弈始终存在唯一的纯策略纳什均衡）中最优响应动态的收敛性。对于同质参与者，我们证明最优响应动态能快速收敛至均衡态。针对两个同质参与者的情况，我们证明了在Θ(log log(1/ϵ))步内收敛至ϵ-近似均衡。当参与者数量n ≥ 3时，由于每步存在n-1 ≥ 2个参与者可进行非平凡行动，动态过程不再唯一。我们考虑Ghosh与Goldberg（2023）提出的随机选择模型——每步随机选取一个参与者执行行动。研究表明，在概率至少为1-δ的条件下，算法将在O(β log(n/(ϵδ)))步内收敛至ϵ-近似均衡，其中β是参与者选择过程的参数（例如当每步均匀随机选择参与者时，β = n² log(n)）。我们补充了适用于任意参与者选择过程的下界结果Ω(n + log(1/ϵ)/log(n))。