Consensus Time in 3-Majority and 2-Choices Is Determined by the Maximum Initial Opinion Density

Abstract

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 $\hat{I}{\pm }^{(0)}$ be the initial opinion-frequency vector, and denote by $\Vert \hat{I}{\pm }^{(0)}\Vert \_\infty$ its maximum entry. We show that 3-Majority reaches consensus in $\tilde{\hat{I}}˜(\min \{\Vert \hat{I}{\pm }^{(0)}\Vert \_{\infty }^{-1},\sqrt{n}\})$ rounds w.h.p., while 2-Choices reaches consensus in $\tilde{\hat{I}}˜(\Vert \hat{I}{\pm }^{(0)}\Vert \_{\infty }^{-1})$ 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 $\Vert \hat{I}{\pm }^{(0)}\Vert \_\infty$, the maximum initial opinion density.