Most social species self-organize into dominance hierarchies(1,2), which decreases aggression and conserves energy(3,4), but it is not clear how individuals know their social rank. We have only begun to learn how the brain represents social rank(5-9) and guides behaviour on the basis of this representation. The medial prefrontal cortex (mPFC) is involved in social dominance in rodents(7,8) and humans(10,11). Yet, precisely how the mPFC encodes relative social rank and which circuits mediate this computation is not known. We developed a social competition assay in which mice compete for rewards, as well as a computer vision tool (AlphaTracker) to track multiple, unmarked animals. A hidden Markov model combined with generalized linear models was able to decode social competition behaviour from mPFC ensemble activity. Population dynamics in the mPFC predicted social rank and competitive success. Finally, we demonstrate that mPFC cells that project to the lateral hypothalamus promote dominance behaviour during reward competition. Thus, we reveal a cortico-hypothalamic circuit by which the mPFC exerts top-down modulation of social dominance.