Analysis of timescale to consensus in voting dynamics with more than two options

We generalize a binary majority-vote model on adaptive networks to its plurality-vote counterpart and analyze the time scale to consensus when voters are given more than two options. When opinions are uniformly distributed in the population of voters in the initial state, we find that the time scale to consensus is shorter than the binary vote model from both numerical simulations and mathematical analysis using the master equation for the three-state plurality-vote model. When intervention such as opinion conversion is allowed, as in the case of sudden change of mind of voter for any reason, the effort needed to push the fragmented three-opinion population in the thermodynamic limit to the consensus state, measured in minimal intervention cost, is less than that needed to push a polarized two-opinion population to the consensus state, when the degree ($p$) of homophily is less than 0.8. For finite system, the fragmented three-opinion population will spontaneously reach the consensus state, with faster time to consensus, compared to polarized two-opinion population, for a broad range of $p$.