Naming game with greater stubbornness and unilateral zealots

We examine a modified naming game on a complete network. In this modification, there are multiple intermediate states rather than one, with the intermediate states evenly spaced between commitment to opinion A and commitment to opinion B. The presence of zealots with unshakable support for opinion A was added to the model. The system tends to a 1-dimensional center manifold, either reaching consensus at A or near-consensus near B. This behavior is similar to the standard naming game on the complete graph. Special attention was paid to the case of two intermediate states. It was found that near-consensus at B could only be reached when the number of zealots was below a certain value. In the case of a single intermediate state, this value is 7.2%, of the total population. This is consistent with the work of Zhang and Lim, who found that the tipping point was 8% ± 1%[1] In the case of two intermediate states, this value was 12.1%.