Allocation of Prizes in Contests with Participation Constraints

We study all-pay contests with an exogenous minimal effort constraint where a player can participate in a contest only if his effort (output) is equal to or higher than the minimal effort constraint. Contestants are privately informed about a parameter (ability) that affects their cost of effort. The designer decides about the size and the number of prizes. We analyze the optimal prize allocation for the contest designer who wishes to maximize either the total effort or the highest effort. It is shown that if the minimal effort constraint is relatively high, the winner-take-all contest in which the contestant with the highest effort wins the entire prize sum does not necessarily maximize the expected total effort nor the expected highest effort. In that case, the random contest in which the entire prize sum is equally allocated to all the participants is a legitimate alternative to the winner-take-all contest.