Punishment in an Open Contest

In the literature on contests, punishments have received much less attention than prizes. One possible reason is that punishing the bottom player(s) in a contest where contestants are not allowed to quit, while e¤ective in increasing contestants’total e¤ort, often violates individual rationality constraints –low ability players may have a negative expected utility in equilibrium since they have to participate. But what will happen in an open contest where all (potential) contestants can choose whether or not to participate? We study a model of this type and allow the contest designer to punish the bottom participant according to their performance. We conclude that punishment is often not desirable (optimal punishment is zero) when the contest designer wants to maximize the expected total e¤ort, while punishment is often desirable (optimal punishment is strictly positive) when the contest designer wants to maximize the expected highest individual e¤ort, i.e., when the contest designer only cares about the performance of the top contestant.