Statistical manipulability of social choice functions

We consider the proportion of profiles at which an anonymous neutral social choice function cannot be manipulated to any coalition's advantage. We fix the number of alternatives and consider the limit as the number of individuals goes to infinity, and assume that the limit of the fraction of all profiles in the domain is 1. We show that there is a continuous function on “profile vectors’ which gives the maximum probability of strategy-proofness, and that this is strictly between 0 and 1 for three alternatives. We also analyze a number of common social choice functions along these lines.