Restricting Entries to All-Pay Contests

We study an all-pay contest where players with low abilities are filtered prior to the round of competing for prizes. These are often practiced due to limited resources or to enhance the competitiveness of the contest. We consider a setting where the designer admits a certain number of top players into the contest. The players admitted into the contest update their beliefs about their opponents based on the signal that their abilities are among the top. We find that their posterior beliefs, even with IID priors, are correlated and depend on players' private abilities, representing a unique feature of this game. We explicitly characterize the symmetric and unique Bayesian equilibrium strategy. We find that each admitted player's equilibrium effort is in general not monotone with the number of admitted players. Despite this non-monotonicity, surprisingly, all players exert their highest efforts when all players are admitted. This result holds generally -- it is true under any ranking-based prize structure, ability distribution, and cost function. We also discuss a two-stage extension where players with top first-stage efforts can proceed to the second stage competing for prizes.