The Conclave Process

We introduce a stochastic model for the papal conclave in which $n$ cardinals vote repeatedly among themselves until one cardinal receives all the votes. In each round, the probability that a cardinal votes for a given candidate is proportional to the $\alpha$-th power of that candidate's vote count in the preceding round. For $\alpha=1$, the model reduces to the Wright-Fisher model and is dual to Kingman's n-coalescent. We reveal a sharp transition in the absorption time $\mathcal{T}$ at $\alpha=1$. It was known that when $\alpha=1$, $\mathcal{T}$ is typically of order $n$. We prove that for $\alpha>1$, it drops to order $\textit{loglog n.}$ In contrast, for $\alpha<1$, $\mathcal{T}$ is typically at least $\exp(\Omega(n))$. We also prove a sharp phase transition in the identity of the winner when $\alpha>1$. For every positive integer $k$, if $2^{1/k}<\alpha<2^{1/(k-1)}$ (where we write $2^{1/0} = +\infty$), with probability tending to 1 as $n\to\infty$, the eventual winner is the unique leader after round $k$. These results show that reinforced voting processes reach consensus remarkably quickly even for large electorates.