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 α^(0) be the initial opinion-frequency vector, and denote by ‖α^(0)‖_∞ its maximum entry. We show that 3-Majority reaches consensus in Θ̃(min{‖α^(0)‖_∞^{-1},√n}) rounds w.h.p., while 2-Choices reaches consensus in Θ̃(‖α^(0)‖_∞^{-1}) rounds w.h.p.
Our results demonstrate that the convergence time of both dynamics is governed not by aggregate statistics such as the number of opinions k or the squared 𝓁₂ norm of the initial opinion distribution, but rather by the extremal statistic ‖α^(0)‖_∞, the maximum initial opinion density.