We examine the choices of male and female students in higher education in a competitive environment. Undergraduate students at a selective French university compete for spots in foreign universities where they will fulfill their mandatory international exchange program requirements. Holding fixed the field of study and accounting for individuals’ underlying academic ability, we find that average- and high-ability female students systematically request universities that are worse than their academic standing. This result is consistent with the “women don’t ask” literature. We then study why women don’t ask, and find some evidence that risk-aversion explains part of female students’ behavior. In a third analysis, we ask investigate whether women would get if they did ask, and find evidence that high-ability female students may have incentives not to ask. Using simple simulations, we consider the impact of other allocation rules on students’ assignments for the international exchange year. We find that both male and female students would be better off on average if female students asked for better-ranked schools. Average male students in particular would be better off. Finally, we study whether women actually want to ask for better-ranked schools. We find differences in students’ preferences for academic experiences, which may explain to a great extent why women don’t ask. While the academic prestige of a university matters to both male and female students, we find evidence that female students also value other university characteristics, such as the cultural environment.