We provide sufficient conditions for equilibrium existence in first‐price auctions with private values that accommodate non quasi‐linear utilities and value‐distributions that contain atoms and exhibit positive or negative correlation. These conditions show that equilibrium existence often turns on properties of a single statistic of the joint distribution of values, namely, the minimum value in the support of the high‐value distribution (the mHV). We also show that modifying the standard tie‐breaking rule only at the mHV is enough to guarantee equilibrium existence without our sufficient conditions. Our results also apply to Bertrand price competition when each firm's constant marginal cost is private information.

Many countries base college admissions on a centrally administered test. Students invest a great deal of resources to improve their performance on the test, and there is growing concern about the high costs associated with these activities. We consider modifying the test by introducing performance-disclosure policies that pool intervals of performance rankings. Pooling affects the equilibrium allocation of students to colleges, which hurts some students and benefits others, but also affects students’ effort. We investigate how such policies can improve students’ welfare in a Pareto sense, study the Pareto frontier of pooling policies, and identify improvements that are robust to the distribution of college seats. We illustrate the potential applicability of our results with an empirical estimation that uses data on college admissions in Turkey. We find that a policy that pools a large fraction of the lowest-performing students leads to a Pareto improvement in a contest based on the estimated parameters. A laboratory experiment based on the estimated parameters generally supports our theoretical predictions.