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

Universal and non-universal features of the generalized voter class for ordering dynamics