A model of voting dynamics under bounded confidence with nonstandard norming

In this paper, we study a model of opinion dynamics based on the so-called "bounded confidence" principle introduced by Hegselmann and Krause. Following this principle, voters participating in an electoral decision with two options are influenced by individuals sharing an opinion similar to their own.

We consider a modification of this model where the operator generating the dynamical system which describes the process of formation the final distribution of opinions in the society is defined in two steps. First, to the opinion of an agent, a value proportional to opinions in his/her "influence group" is added, and then the elements of the resulting array are divided by the maximal absolute value of elements to keep the opinions in the prescribed interval. We show that under appropriate conditions, any trajectory tends to a fixed point, and all the remaining fixed points are Lyapunov stable.