The optimal allocation of prizes in contests with costly entry

Contestants often need to incur an opportunity cost to participate in the competition. In this paper, we accommodate costly entry and study the effort-maximizing prize allocation rule in a contest environment of all-pay auction with incomplete information as in Moldovanu and Sela (2001). As equilibrium entry can be stochastic, our analysis allows prize allocation rule to be contingent on the number of entrants. With free entry, Moldovanu and Sela establish the optimality of winner-take-all when effort cost function is linear or concave. Costly entry introduces a new trade-off between eliciting effort from entrants and encouraging entry of contestants, which might demand a more lenient optimal prize allocation rule. Surprisingly, we find that the optimality of winner-take-all is robust to costly entry when cost function is linear or concave. On the other hand, we provide examples to show that the new trade-off due to costly entry does make a difference to the optimal design when effort cost function is convex. (c) 2019 Elsevier B.V. All rights reserved.