  In this paper, we propose a framework to study a general class of strategic
behavior in voting, which we call vote operations. We prove the following
theorem: if we fix the number of alternatives, generate $n$ votes i.i.d.
according to a distribution $\pi$, and let $n$ go to infinity, then for any
$\epsilon >0$, with probability at least $1-\epsilon$, the minimum number of
operations that are needed for the strategic individual to achieve her goal
falls into one of the following four categories: (1) 0, (2) $\Theta(\sqrt n)$,
(3) $\Theta(n)$, and (4) $\infty$. This theorem holds for any set of vote
operations, any individual vote distribution $\pi$, and any integer generalized
scoring rule, which includes (but is not limited to) almost all commonly
studied voting rules, e.g., approval voting, all positional scoring rules
(including Borda, plurality, and veto), plurality with runoff, Bucklin,
Copeland, maximin, STV, and ranked pairs.
  We also show that many well-studied types of strategic behavior fall under
our framework, including (but not limited to) constructive/destructive
manipulation, bribery, and control by adding/deleting votes, margin of victory,
and minimum manipulation coalition size. Therefore, our main theorem naturally
applies to these problems.

Artificial Intelligence;Computer Science and Game Theory