Best-Response Dynamics in Lottery Contests

We study the convergence of best-response dynamics in lottery contests. We show that best-response dynamics rapidly converges to the (unique) equilibrium for homogeneous agents but may not converge for non-homogeneous agents, even for two non-homogeneous agents. For $2$ 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 a model 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., $\beta = n$ if agents are selected uniformly at random at each time step. Our simulations indicate that this bound is tight.