  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 ...
