  We study the fixation time of the identity of the leader, i.e., the most
massive component, in the general setting of Aldous's multiplicative coalescent
[4, 5], which in an asymptotic sense describes the evolution of the component
sizes of a wide array of near-critical coalescent processes, including the
classical Erd\H{o}s-R\'enyi process.
  We show tightness of the fixation time in the "Brownian" regime, explicitly
determining the median value of the fixation time to within an optimal $O(1)$
window. This generalizes {\L}uczak's result [31] for the Erd\H{o}s-R\'enyi
random graph using completely different techniques.
  In the heavy-tailed case, in which the limit of the component sizes can be
encoded using a thinned pure-jump L\'{e}vy process, we prove that only
one-sided tightness holds. This shows a genuine difference in the possible
behavior in the two regimes.
  The solution to the leader problem in the setting of the Erd\H{o}s-R\'enyi
random graph played an important role in the study of the scaling limit of the
minimal spanning tree on the complete graph [2]. We believe that analogous
results, such as those proved herein, will be useful in establishing
universality of the intrinsic geometry of the minimal spanning tree across a
large class of models.
