We consider a general class of imperfectly discriminating contests with privately informed players. We show that findings by Athey (2001) imply the existence of a Bayesian Nash equilibrium in monotone pure strategies.

We consider a two-player all-pay auction with symmetric independent private values that are uniformly distributed. The designer chooses the size of a head start that is given to one of the players. The designer’s objective is to maximize a convex combination of the expected highest effort and...