BestResponse Dynamics in Tullock Contests with Convex Costs
Abstract
We study the convergence of bestresponse dynamics in Tullock contests with convex cost functions (these games always have a unique purestrategy Nash equilibrium). We show that bestresponse 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 $n1 \ge 2$ agents can make nontrivial 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2023
 DOI:
 10.48550/arXiv.2310.03528
 arXiv:
 arXiv:2310.03528
 Bibcode:
 2023arXiv231003528G
 Keywords:

 Computer Science  Computer Science and Game Theory;
 Economics  Theoretical Economics
 EPrint:
 43 pages. WINE '23 version