  Do runoff elections, using the same voting rule as the initial election but
just on the winning candidates, increase or decrease the complexity of
manipulation? Does allowing revoting in the runoff increase or decrease the
complexity relative to just having a runoff without revoting? For both weighted
and unweighted voting, we show that even for election systems with simple
winner problems the complexity of manipulation, manipulation with runoffs, and
manipulation with revoting runoffs are independent, in the abstract. On the
other hand, for some important, well-known election systems we determine what
holds for each of these cases. For no such systems do we find runoffs lowering
complexity, and for some we find that runoffs raise complexity. Ours is the
first paper to show that for natural, unweighted election systems, runoffs can
increase the manipulation complexity.
