Gibbard [gibbard, a., 1973. Manipulation of voting schemes: a general result. Econometrica 41, 587–602] and satterthwaite [satterthwaite, m., 1975. Strategy-proofness and arrow's conditions: existence and correspondence theorems for voting procedures and social welfare functions. Journal of economic theory 10,187–217] show that an anonymous social choice function with more than two alternatives in its range must be manipulable. Under the constraint that the number of agents is larger than the number of alternatives if the latter is four, and larger than this number plus one if it is at least five, we derive the lower bound on the number of manipulable profiles of such social choice functions. Moreover, all such social choice functions attaining this lower bound are characterized. These social choice functions exhibit a trade off between minimizing manipulability and treating alternatives neutrally.
Maus, S., Peters, H. J. M., & Storcken, A. J. A. (2007). Minimally Manipulable Anonymous Social Choice Functions. Mathematical Social Sciences, 53, 239-254. https://doi.org/10.1016/j.mathsocsci.2007.02.002