  By considering three different spin models belonging to the generalized voter
class for ordering dynamics in two dimensions [I. Dornic, \textit{et al.} Phys.
Rev. Lett. \textbf{87}, 045701 (2001)], we show that they behave differently
from the linear voter model when the initial configuration is an unbalanced
mixture up and down spins. In particular we show that for nonlinear voter
models the exit probability (probability to end with all spins up when starting
with an initial fraction $x$ of them) assumes a nontrivial shape. The change is
traced back to the strong nonconservation of the average magnetization during
the early stages of dynamics. Also the time needed to reach the final consensus
state $T_N(x)$ has an anomalous nonuniversal dependence on $x$.
