The Conclave Process

Abstract

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 $\text{loglog n.}$ 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.