A Quantitative Version of the Gibbard-Satterthwaite Theorem for Three   Alternatives
  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. 
Ehud Friedgut, Gil Kalai, Nathan Keller, and Noam Nisan
math.CO cs.AI math.PR