Higher education--particularly the research-intensive university, which is the focus of this article--is the subject of global/national/local effects, and is shaped by hierarchy and uneven development on a world scale. The article theorises social competition in higher education, and traces inter-university competition and stratification on the national and global planes with the help of figures and tables. It argues that social competition is much broader than economic exchange, but in the neo-liberal era marketisation is becoming more important, particularly cross-border markets. Globalisation and markets together are changing the competition for status goods (positional goods) in higher education. The competition is becoming more "economised" because mediated by private capacity to pay, and intensified because there is diminished attention to public good objectives such as equality of opportunity: in any case transnational markets are configured as a trading environment where such objectives are irrelevant. The outcome is the steepening of university hierarchies, the formation of a "winner-take-all" world market in elite and mostly American university education, a tighter fit between social hierarchy and educational hierarchy at the national level, and global patterns of domination/subordination that are as yet scarcely modified by global public goods. This suggests the need to rework the equality of the educational project and situate it globally as well as nationally. (Contains 7 figures, 6 tables, and 39 notes.)