Fixation of leadership in non-Markovian growth processes

TL;DR

This paper studies the long-term leadership behavior in a model of multiple agents with non-Markovian growth, providing conditions for dominance, monopoly, and the emergence of a single leader.

Contribution

It introduces a novel non-Markovian growth model allowing general waiting times, and establishes criteria for leadership, monopoly, and convergence properties, extending previous exponential-based results.

Findings

Leadership occurs with probability zero or one under certain conditions.

Criteria for a single agent to dominate or monopolize are established.

Results include conditions for convergence and the presence of atoms in distributions.

Abstract

Consider a model where $N$ equal agents possess `values', belonging to $\mathbb{N}_0$, that are subject to incremental growth over time. More precisely, the values of the agents are represented by $N$ independent, increasing $\mathbb{N}_0$ valued processes with random, independent waiting times between jumps. We show that the event that a single agent possesses the maximum value for all sufficiently large values of time (called `leadership') occurs with probability zero or one, and provide necessary and sufficient conditions for this to occur. Under mild conditions we also provide criteria for a single agent to become the unique agent of maximum value for all sufficiently large times, and also conditions for the emergence of a unique agent having value that tends to infinity before `explosion' occurs (i.e. conditions for `strict leadership' or `monopoly' to occur almost surely). The novelty of this model lies in allowing non-exponentially distributed waiting times between jumps in value. In the particular case when waiting times are mixtures of exponential distributions, we improve a well-established result on the `balls in bins' model with feedback, removing the requirement that the feedback function be bounded from below and also allowing random feedback functions. As part of the proofs we derive necessary and sufficient conditions for the distribution of a convergent series of independent random variables to have an atom on the real line, a result which we believe may be of interest in its own right.