  Single-peakedness is one of the most important and well-known domain
restrictions on preferences. The computational study of single-peaked
electorates has largely been restricted to elections with tie-free votes, and
recent work that studies the computational complexity of manipulative attacks
for single-peaked elections for votes with ties has been restricted to
nonstandard models of single-peaked preferences for top orders. We study the
computational complexity of manipulation for votes with ties for the standard
model of single-peaked preferences and for single-plateaued preferences. We
show that these models avoid the anomalous complexity behavior exhibited by the
other models. We also state a surprising result on the relation between the
societal axis and the complexity of manipulation for single-peaked preferences.
