Probability laws of consensus in a broadcast-based consensus-forming algorithm

Abstract The consensus attained in the consensus-forming algorithm is not generally a constant but rather a random variable, even if the initial opinions are the same. In the present paper, we investigate the probability laws of the consensus in a broadcast-based consensus-forming algorithm. First, we derive a fundamental equation on the time evolution of the opinions of agents. From the derived equation, we show that the consensus attained by the algorithm is given as a fixed-point solution of a linear equation. We then focus on two extreme cases: consensus forming by two agents and consensus forming by an infinite number of agents. In the two-agent case, we derive several properties of the distribution function of the consensus with an algorithm for computing the distribution function of the consensus numerically. In the infinite-number-of-agents case, we show that if the initial opinions follow a stable distribution, then the consensus also follows a stable distribution. In addition, we derive a closed-form expression of the probability density function of the consensus when the initial opinions follow a Gaussian distribution, a Cauchy distribution, or a Lévy distribution.