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Galam's bottom-up hierarchical system and public debate model revisited

This article is concerned with the bottom-up hierarchical system and public debate model proposed by Galam, as well as a spatial version of the public debate model. In all three models, there is a population of individuals who are characterized by one of two competing opinions, say opinion -1 and opinion +1. This population is further divided into groups of common size s. In the bottom-up hierarchical system, each group elects a representative candidate, whereas in the other two models, all the members of each group discuss at random times until they reach a consensus. At each election/discussion, the winning opinion is chosen according to Galam's majority rule: the opinion with the majority of representants wins when there is a strict majority while one opinion, say opinion -1, is chosen by default in case of a tie. For the public debate models, we also consider the following natural updating rule that we shall call proportional rule: the winning opinion is chosen at random with a probability equal to the fraction of its supporters in the group. The three models differ in term of their population structure: in the bottom-up hierarchical system, individuals are located on a finite regular tree, in the non-spatial public debate model, they are located on a complete graph, and in the spatial public debate model, they are located on the $d$-dimensional regular lattice. For the bottom-up hierarchical system and non-spatial public debate model, Galam studied the probability that a given opinion wins under the majority rule and assuming that individuals' opinions are initially independent, making the initial number of supporters of a given opinion a binomial random variable. The first objective of this paper is to revisit his result assuming that the initial number of individuals in favor of a given opinion is a fixed deterministic number ...