In strategyproof mechanisms, it is always an optimal strategy for agents to truthfully report their private information to the mechanism. This theoretical property is clearly appealing, as it gives a mechanism designer the ability to predict play and make meaningful statements about other criteria such as welfare. However, a growing body of empirical evidence has documented significant deviations from truthful reporting in such mechanisms. This issue is particularly important in matching markets, the focus in this paper.

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 Impossibililty 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 an agent desires---those that they have been rejected by---should be ranked higher than the one they receive. We provide a mechanism to produce a social ranking given any preference profile and outcome assignment, and characterize this ranking as the unique one that satisfies certain desirable axioms.