  The Gibbard-Satterthwaite theorem states that every non-dictatorial election
rule among at least three alternatives can be strategically manipulated. We
prove a quantitative version of the Gibbard-Satterthwaite theorem: a random
manipulation by a single random voter will succeed with a non-negligible
probability for any election rule among three alternatives that is far from
being a dictatorship and from having only two alternatives in its range.
