Solution of the multi-state voter model and application to strong neutrals in the naming game

We consider the voter model with $M$ states initially in the system. Using generating functions, we pose the spectral problem for the Markov transition matrix and solve for all eigenvalues and eigenvectors exactly. With this solution, we can find all future probability probability distributions, the expected time for the system to condense from $M$ states to $M-1$ states, the moments of consensus time, the expected local times, and the expected number of states over time. Furthermore, when the initial distribution is uniform, such as when $M=N$, we can find simplified expressions for these quantities. In particular, we show that the mean and variance of consensus time for $M=N$ is $\frac{1}{N}(N-1{)}^2$ and $\frac{1}{3}({\pi }^2-9)(N-1{)}^2$ respectively.