This paper introduces the q-fixed majority property for committee selection rules, which extends the traditional fixed majority principle to a flexible framework. We examine conditions under which the committee scoring rules satisfy the q-fixed majority property. Focusing on (weakly) separable rules, we find that the Bloc rule is the only which satisfies it for all q > 1/2. In addition, the q-bottom majority property is introduced, highlighting conditions under which committees can be excluded based on voter consensus.
