 Geometric continuation after an initial tie

Suppose a contest uses $N_0$ trials initially and ends there with probability $1-\alpha$. Conditional on an initial tie, independent further rounds each use $\ell$ trials and resolve the contest with a fixed probability $0<r\le 1$. The number $J$ of further rounds, conditional on that tie, has a geometric distribution: $\mathrm{Pr}(J=j\mid \text{tie})=(1-r{)}^{j-1}r$ for $j\ge 1$. Thus the total number $N$ of trials has an atom $1-\alpha$ at $N_0$ and the displayed probabilities thereafter. The geometric series sums to $\alpha$, so termination has probability one, and the expected value is $\mathbb{E}[N]=N_0+\ell \alpha /r$. The displayed distribution requires independence of the future rounds from the initial tie.