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 \cite{ghosh2023best}, 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：我们研究了具有凸成本函数的Tullock竞赛中最佳响应动力学的收敛性（此类博弈始终存在唯一的纯策略纳什均衡）。研究表明，对于同质参与者，最佳响应动力学能快速收敛至均衡。对于两个同质参与者，我们证明了在$\Theta(\log\log(1/\epsilon))$步内可收敛至$\epsilon$-近似均衡。当参与者数量$n \ge 3$时，动力学过程并非唯一，因为每一步中$n-1 \ge 2$个参与者可能做出非平凡移动。我们考虑了\cite{ghosh2023best}提出的模型，其中每一步随机选择移动参与者。我们证明，以概率$1-\delta$，在$O(\beta \log(n/(\epsilon\delta)))$步内可收敛至$\epsilon$-近似均衡，其中$\beta$为参与者选择过程的参数，例如当每一步均匀随机选择参与者时$\beta = n^2 \log(n)$。我们通过适用于任意参与者选择过程的下界$\Omega(n + \log(1/\epsilon)/\log(n))$补充了这一结果。