In voting problems where agents have lipschitz continuous utility functions on a multidimensional space of alternatives, a voting rule is threshold strategy-proof if any agent can obtain only a limited utility gain by not voting for a most preferred alternative, if the number of agents is large enough. For anonymous voting rules it is shown that this condition is not only implied by but is in fact equivalent to the influence of any single agent decreasing to zero as the number of agents grows. If there are at least five agents, the mean rule (taking the average vote) is shown to be the unique anonymous and unanimous voting rule that meets a lower bound with respect to the number of agents needed to obtain threshold strategy-proofness.
Peters, H. J. M., Ehlers, L. H., & Storcken, A. J. A. (2004). Threshold Strategy-Proofness: On Manipulability in Large Voting Systems. Games and Economic Behavior, 49, 103-116. https://doi.org/10.1016/j.geb.2003.10.004