We consider the problem of aggregating individual preferences over alternatives into a social ranking. A key feature of the problems that we consider—and the one that allows us to obtain positive results, in contrast to negative results such as Arrow’s Impossibility Theorem—is that the alternatives to be ranked are outcomes of a competitive process. Examples include rankings of colleges or academic journals. The foundation of our ranking method is that alternatives that agents rank higher than the one they receive (and thus have been rejected by) should also be ranked higher in the aggregate ranking. We introduce axioms to formalize this idea, and call any ranking that satisfies our axioms a desirable ranking. We show that as the market grows large, any desirable ranking coincides with the true underlying ranking of colleges by quality. Last, we provide an algorithm for constructing desirable rankings, and show that the outcome of this algorithm is the unique ranking of the colleges that satisfy our axioms. Monte Carlo experiments show that our proposed desirable ranking perform well even when market is moderately large.