Fast Convergence of $k$-Opinion Undecided State Dynamics in the Population Protocol Model

We analyze the convergence of the $k$-opinion Undecided State Dynamics (USD) in the population protocol model. For $k$=2 opinions it is well known that the USD reaches consensus with high probability within $O(n \log n)$ interactions. Proving that the process also quickly solves the consensus problem for $k>2$ opinions has remained open, despite analogous results for larger $k$ in the related parallel gossip model. In this paper we prove such convergence: under mild assumptions on $k$ and on the initial number of undecided agents we prove that the USD achieves plurality consensus within $O(k n \log n)$ interactions with high probability, regardless of the initial bias. Moreover, if there is an initial additive bias of at least $\Omega(\sqrt{n} \log n)$ we prove that the initial plurality opinion wins with high probability, and if there is a multiplicative bias the convergence time is further improved. Note that this is the first result for $k > 2$ for the USD in the population protocol model. Furthermore, it is the first result for the unsynchronized variant of the USD with $k>2$ which does not need any initial bias.