We establish the correct parameter governing the convergence time of the 3-Majority and 2-Choices dynamics on the complete graph in the synchronous model. Recent work [Shimizu and Shiraga, PODC'25] provides matching upper and lower bounds on the number of rounds to consensus, but only in a weak sense: the bounds are shown to coincide for some initial opinion configuration. In contrast, we obtain tight bounds in a strong sense, with upper and lower bounds matching up to logarithmic factors for every initial configuration. Let $α$ (0) be the initial opinion-frequency vector, and denote by ___$α$ (0) ___ $\infty$ its maximum entry. We show that 3-Majority reaches consensus in $Θ$(min{___$α$ (0) ___ -1 $\infty$ , $\sqrt$ n}) rounds w.h.p., while 2-Choices reaches consensus in $Θ$(___$α$ (0) ___ -1 $\infty$ ) rounds w.h.p. Our results demonstrate that the convergence time of both dynamics is governed not by global parameters such as the number of opinions k or the squared ${\ell}$ 2 norm of the initial opinion distribution, but rather by the ''local'' parameter ___$α$ (0) ___ $\infty$ , the maximum initial opinion density.

翻译：我们建立了同步模型下完全图上3-多数决（3-Majority）与二选一（2-Choices）动力学收敛时间的正确控制参数。近期研究[Shimizu与Shiraga，PODC'25]给出了达成共识所需轮次的匹配上下界，但仅局限于弱意义：这些界仅在特定初始观点配置下一致。相比之下，我们在强意义下获得了紧界，使得每个初始配置的上下界在对数因子范围内匹配。设$\alpha(0)$为初始观点频率向量，记$\|\alpha(0)\|_\infty$为其最大分量。我们证明，3-多数决以高概率在$\Theta(\min\{\|\alpha(0)\|_\infty^{-1}, \sqrt{n}\})$轮内达成共识，而二选一则以高概率在$\Theta(\|\alpha(0)\|_\infty^{-1})$轮内达成共识。我们的结果表明，两种动力学的收敛时间并非由全局参数（如观点数量k或初始观点分布的$\ell_2$范数平方）主导，而是由"局部"参数$\|\alpha(0)\|_\infty$（即最大初始观点密度）决定。