Competition involves two dimensions, rivalry for resources and status-ranking. Our experiment isolates the effects of the latter. Participants do a task under non-rivalry incentives. Before doing so, individuals indicate whether they choose an environment with social-status ranking or one without. When a man does a ranking that is imposed on all others, women choose status-ranking less frequently than men. There is no gender difference with a female ranker. This finding complements the established result that women are averse to competing under rivalry for resources, in a qualified way. Women also exhibit status-ranking aversion, but only when ranked by a man.