A model of anonymous influence with anti-conformist agents

Abstract

We study a stochastic model of anonymous influence with conformist and anti-conformist individuals. Each agent with a 'yes' or 'no' initial opinion on a certain issue can change his opinion due to social influence. We consider anonymous influence, which depends on the number of agents having a certain opinion, but not on their identity. An individual is conformist/anti-conformist if his probability of saying 'yes' increases/decreases with the number of 'yes'-agents. We focus on three classes of aggregation rules (pure conformism, pure anti-conformism, and mixed aggregation rules) and examine two types of society (without, and with mixed agents). For both types we provide a complete qualitative analysis of convergence, i.e., identify all absorbing classes and conditions for their occurrence. Also the pure case with infinitely many individuals is studied. We show that, as expected, the presence of anti-conformists in a society brings polarization and instability: polarization in two groups, fuzzy polarization (i.e., with blurred frontiers), cycles, periodic classes, as well as more or less chaotic situations where at any time step the set of 'yes'-agents can be any subset of the society. Surprisingly, the presence of anti-conformists may also lead to opinion reversal: a majority group of conformists with a stable opinion can evolve by a cascade phenomenon towards the opposite opinion, and remains in this state.JEL Classification: C7, D7, D85

Anti-conformism in the Threshold Model of Collective Behavior

“…Our paper is in the line of a previous work by the first author [13], whose results will be used at some point in the present paper. Our study bears essentially in answering the main question raised in the first paragraph, that is, on the convergence of the process, analyzing if absorbing states exist (stable state of the society) or if a cycle occurs, or even more chaotic situations.…”

“…is a nonincreasing function of s, starting at 1 with s = 0 and finishing at 0 with s = n. Thus, we fall into the framework studied in Grabisch et al [13] on an anonymous model of anti-conformism where each agent i has the probability p i (s) to take action 1 at next step knowing that the current state is S, with s = |S|, and p i (s) is a nondecreasing (respectively, nonincreasing) function reaching values 0 and 1 when i is conformist (respectively, anti-conformist).…”

“…In the model of Grabisch et al [13], all functions p i can be different among the agents, but it is required that for all conformists, the functions p i have the same domain where they take value 0 and 1, and similarly for the anti-conformists. These domains are characterized for the conformists by the quantities l c (firing threshold) and n − r c (saturation threshold) which are the rightmost and leftmost values of s for which p 1 i (s) is 0 and 1, respectively, given by…”

“…From Grabisch et al [13], we know that in full generality 20 possible absorbing classes can occur, depending on the values of l c , r c , l a , r a . Since it holds that in our case l c + r c = l a + r a = d − 1 < n − 1, 5 among the 20 are not possible.…”

“…We examine different typical situations for the value of the parameters, taking advantage of the study made in Grabisch et al [13] (full detail can be found in this reference).…”