2026/07/24 by Itai Benjamini, Zhenhao Cai, Guanyi Chen +2
#math.PR #math.CO
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 α-th power of that candidate's vote count in the preceding round. For α=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 T at α=1. It was known that when α=1, T is typically of order n. We prove that for α>1, it drops to order loglog n. In contrast, for α<1, T is typically at least exp(Ω(n)). We also prove a sharp phase transition in the identity of the winner when α>1. For every positive integer k, if 21/k<α<21/(k-1) (where we write 21/0 = +∞), with probability tending to 1 as n→∞, 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.