Optimal punishment in contests with endogenous entry

We study optimal punishment in an all-pay contest with endogenous entry, where the participant with the lowest performance may be punished. When a small punishment is introduced, the lowest ability players drop out and those of medium ability exert less effort, while only the highest ability players exert more effort. A sufficient condition is given for the optimal punishment to be zero if the objective is to maximize the expected total effort. As cost functions become more convex, punishment becomes less desirable. When the objective is to maximize the expected highest individual effort, a positive punishment is desirable under much weaker conditions.