A new supplement to McGarvey's wellknown theorem is given. Namely, we prove
that for each lambda with 1/2 < lambda < 1 there exists a tournament T on
some finite set of alternatives A such that for every profile of A there ex
ists an are (a, b) of T such that the proportion of voters that prefer to b
is less than lambda. In other words, there exist tournaments that cannot b
e a lambda-majority relation of any profile. Lower and upper bounds for the
minimal majority with which we can generate all tournaments on n alternati
ves are also given. (C) 1999 Elsevier Science BN. All rights reserved.