How Many Vote Operations Are Needed to Manipulate A Voting System?
  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. 
Lirong Xia
cs.AI cs.GT