When and why do women gain from increased descriptive representation in deliberating bodies? Using a large randomized experiment, and linking individual-level speech with assessments of speaker authority, we find that decision rules interact with the number of women in the group to shape the conversation dynamics and deliberative authority, an important form of influence. With majority rule and few women, women experience a negative balance of interruptions when speaking, and these women then lose influence in their own eyes and in others’. But when the group is assigned to unanimous rule, or when women are many, women experience a positive balance of interruptions, mitigating the deleterious effect of small numbers. Men do not experience this pattern. We draw implications for a type of representation that we call authoritative representation, and for democratic deliberation.