= Geometric continuation after an initial tie
{title2=$\Pr(N=N_0+\ell j)=\alpha(1-r)^{j-1}r$}
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 $0<r\leq1$. The number $J$ of further rounds, conditional on that tie, has a <geometric distribution>: $\Pr(J=j\mid\text{tie})=(1-r)^{j-1}r$ for $j\geq1$. 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.
Back to article page
