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.

翻译：我们分析了种群协议模型中$k$意见待定状态动力学(USD)的收敛性。对于$k=2$种意见，众所周知USD能以高概率在$O(n \log n)$次交互内达成共识。尽管在相关的并行八卦模型中针对较大$k$有类似结果，但证明该过程也能快速解决$k>2$意见的共识问题此前仍未解决。本文证明了此类收敛性：在对$k$和初始待定代理数量做出温和假设的条件下，我们证明USD能以高概率在$O(k n \log n)$次交互内达成多数共识，且与初始偏好无关。此外，若初始存在至少$\Omega(\sqrt{n} \log n)$的加性偏好，我们证明初始多数意见能以高概率胜出；若存在乘性偏好，收敛时间将进一步提升。需注意，这是种群协议模型中USD在$k>2$情况下的首个结果，也是无需任何初始偏好的$k>2$非同步变体USD的首个结果。