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.Comment: 13 pages, 12 figure