On λ-majority voting paradoxes

Abstract A new supplement to McGarvey's wellknown theorem is given. Namely, we prove that for each λ with 1/2 λ T on some finite set of alternatives A such that for every profile of A there exists an arc ( a , b ) of T such that the proportion of voters that prefer a to b is less than λ . In other words, there exist tournaments that cannot be a λ -majority relation of any profile. Lower and upper bounds for the minimal majority with which we can generate all tournaments on n alternatives are also given.