Divide-and-rule policy in the Naming Game

The Naming Game is a classic model for studying the emergence and evolution of language in a population. In this paper, we consider the Naming Game with multiple committed opinions and investigate the dynamics of the game on a complete graph with an arbitrary large population. The homogeneous mixing condition enables us to use mean-field theory to analyze the opinion evolution of the system. However, when the number of opinions increases, the number of variables describing the system grows exponentially. We focus on a special scenario where the largest group of committed agents competes with a motley of committed groups, each of which is significantly smaller than the largest one, while the majority of uncommitted agents initially hold one unique opinion. We choose this scenario for two reasons. The first is that it arose many times in different societies, while the second is that its complexity can be reduced by merging all agents of small committed groups into a single committed group. We show that the phase transition occurs when the group of the largest committed fraction dominates the system, and the threshold for the size of the dominant group at which this transition occurs depends on the size of the committed group of the unified category. Further, we derive the general formula for the multi-opinion evolution using a recursive approach. Finally, we use agent-based simulations to reveal the opinion evolution in the random graphs. Our results provide insights into the conditions under which the dominant opinion emerges in a population and the factors that influence this process.