  We investigate an averaging process that describes how interacting agents
approach consensus through binary interactions. In each elementary step, two
agents are selected at random and they reach compromise by adopting their
opinion average. We show that the fraction of agents with a monotonically
decreasing opinion decays as $e^{-\alpha t}$, and that the exponent
$\alpha=\tfrac{1}{2}-\tfrac{1+\ln \ln 2}{4\ln 2}$ is selected as the extremum
from a continuous spectrum of possible values. The opinion distribution of
monotonic agents is asymmetric, and it becomes self-similar at large times.
Furthermore, the tails of the opinion distribution are algebraic, and they are
characterized by two distinct and nontrivial exponents. We also explore
statistical properties of agents with an opinion strictly above average.
