  Recently, quantitative versions of the Gibbard-Satterthwaite theorem were
proven for $k=3$ alternatives by Friedgut, Kalai, Keller and Nisan and for
neutral functions on $k \geq 4$ alternatives by Isaksson, Kindler and Mossel.
  We prove a quantitative version of the Gibbard-Satterthwaite theorem for
general social choice functions for any number $k \geq 3$ of alternatives. In
particular we show that for a social choice function $f$ on $k \geq 3$
alternatives and $n$ voters, which is $\epsilon$-far from the family of
nonmanipulable functions, a uniformly chosen voter profile is manipulable with
probability at least inverse polynomial in $n$, $k$, and $\epsilon^{-1}$.
  Removing the neutrality assumption of previous theorems is important for
multiple reasons. For one, it is known that there is a conflict between
anonymity and neutrality, and since most common voting rules are anonymous,
they cannot always be neutral. Second, virtual elections are used in many
applications in artificial intelligence, where there are often restrictions on
the outcome of the election, and so neutrality is not a natural assumption in
these situations.
  Ours is a unified proof which in particular covers all previous cases
established before. The proof crucially uses reverse hypercontractivity in
addition to several ideas from the two previous proofs. Much of the work is
devoted to understanding functions of a single voter, and in particular we also
prove a quantitative Gibbard-Satterthwaite theorem for one voter.
