Best-Response Dynamics in Tullock Contests with Convex Costs

TL;DR

This paper analyzes the convergence properties of best-response dynamics in Tullock contests with convex costs, demonstrating rapid convergence for homogeneous agents and probabilistic convergence bounds for multiple agents, with implications for equilibrium computation.

Contribution

It provides new convergence rate results for best-response dynamics in Tullock contests with convex costs, including probabilistic bounds for multiple agents and lower bounds applicable to any agent selection process.

Findings

Rapid convergence for two homogeneous agents.

Probabilistic convergence bounds for n ≥ 3 agents.

Lower bounds on convergence time for any agent selection process.

Abstract

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.