The limit point in the Jante's law process has an absolutely continuous distribution

TL;DR

This paper proves that in a multidimensional opinion consensus model, the final opinion distribution is continuous, confirming a conjecture about the nature of the limit point in the Jante's law process.

Contribution

It establishes that the limit point in the Jante's law process has an absolutely continuous distribution, confirming a key conjecture in the model's analysis.

Findings

The limit opinion distribution is absolutely continuous.

The model's limit point distribution is proven to be continuous.

The result confirms the conjecture made in prior work.

Abstract

We study a stochastic model of consensus formation, introduced in 2015 by Grinfeld, Volkov and Wade, who called it a multidimensional randomized Keynesian beauty contest. The model was generalized by Kennerberg and Volkov, who called their generalization the Jante's law process. We consider a version of the model where the space of possible opinions is a convex body $\mathcal{B}$ in $\mathbb{R}^d$. $N$ individuals in a population each hold a (multidimensional) opinion in $\mathcal{B}$. Repeatedly, the individual whose opinion is furthest from the center of mass of the $N$ current opinions chooses a new opinion, sampled uniformly at random from $\mathcal{B}$. Kennerberg and Volkov showed that the set of opinions that are not furthest from the center of mass converges to a random limit point. We show that the distribution of the limit opinion is continuous, thus proving the conjecture made after Proposition 3.2 in Grinfeld et al.