  We study a family of opinion formation models in one dimension where the
propensity for a voter to align with its local environment depends non-linearly
on the fraction of disagreeing neighbors. Depending on this non-linearity in
the voting rule, the population may exhibit a bias toward zero magnetization or
toward consensus, and the average magnetization is generally not conserved. We
use a decoupling approximation to truncate the equation hierarchy for
multi-point spin correlations and thereby derive the probability to reach a
final state of a given consensus as a function of the initial magnetization.
The case when voters are influenced by more distant voters is also considered
by investigating the Sznajd model.
