  In multiagent settings where the agents have different preferences,
preference aggregation is a central issue. Voting is a general method for
preference aggregation, but seminal results have shown that all general voting
protocols are manipulable. One could try to avoid manipulation by using voting
protocols where determining a beneficial manipulation is hard computationally.
The complexity of manipulating realistic elections where the number of
candidates is a small constant was recently studied (Conitzer 2002), but the
emphasis was on the question of whether or not a protocol becomes hard to
manipulate for some constant number of candidates. That work, in many cases,
left open the question: How many candidates are needed to make elections hard
to manipulate? This is a crucial question when comparing the relative
manipulability of different voting protocols. In this paper we answer that
question for the voting protocols of the earlier study: plurality, Borda, STV,
Copeland, maximin, regular cup, and randomized cup. We also answer that
question for two voting protocols for which no results on the complexity of
manipulation have been derived before: veto and plurality with runoff. It turns
out that the voting protocols under study become hard to manipulate at 3
candidates, 4 candidates, 7 candidates, or never.
