We examine gender differences in competitiveness, using a TV game show where the winner of an elimination competition plays a game of chance worth hundreds of thousands of euros. At several stages of this high-stakes competition, contestants face a choice between continuing to compete and opting out in exchange for a comparatively modest prize. When strategic considerations are absent, we observe the well-known pattern that women are less likely to compete than men, but this difference derives entirely from women avoiding competition against men. When the decision to compete is strategic and contestants should factor in the competitiveness of others, women again avoid competing against men. Men, in turn, seem to anticipate the lower competitiveness of female opponents, as evidenced by their greater tendency to compete against women in such a situation. Ability differences are unlikely to explain these results. The findings underline the importance of the gender of competitors for the analysis of differences in willingness to compete, and shed new light on the persistent gender gap at the male-dominated higher rungs of the career ladder.