# Contests with endogenous entry

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## Abstract

This paper studies the effort-maximizing design of a complete-information contest with endogenous entry. A fixed pool of homogenous potential players with identical marginal bidding cost must incur an entry cost to enter the contest before they bid for prize(s). The designer can flexibly adjust the impact function of a generalized nested lottery contest and use a fixed budget to fund single or multiple prizes. Applying Dasgupta and Maskin (Rev Econ Stud 53(1):1–26, 1986), we establish the existence of symmetric equilibrium for all contest mechanisms concerned. A uniform upper bound for expected overall bids is identified for any eligible contest, assuming that potential bidders play symmetric equilibria. We show that the upper bound can be achieved through a Tullock contest with a single contingent prize, which adopts compatible bundles of success function and entry fees/subsidies. In particular, we identify the conditions under which the optimum can be achieved by solely setting the right discriminatory power in a Tullock contest with a single fixed prize. Finally, our analysis characterizes the optimal shortlisting rule, which reveals that the contest designer generally should exclude potential bidders to elicit higher bids.

### Keywords

Contest design Endogenous entry Entry cost Stochastic entry### JEL Classification

C7 D8## Notes

### Acknowledgments

We are grateful to the Co-Editor in charge Vijay Krishna, the associate editor and two anonymous referees for very insightful and constructive comments and suggestions. We thank Atsu Amegashie, Masaki Aoyagi, Helmut Bester, Oliver Gürtler, Todd Kaplan, Dan Kovenock, John Morgan, Johannes Münster, Ella Segev, Aner Sela, Roman Sheremeta, Randy Silvers, Ching-jen Sun, Samarth Vaidya, Cédric Wasser, and Elmar Wolfstetter, participants in the 2010 SAET Meetings, the 2011 International Conference on Contests, Tournaments and Relative Performance Evaluation, the 2011 Young Researcher Workshop on Contests and Tournaments (Berlin), and American Economic Association Annual Meetings 2012, as well as seminar participants at Deakin University and the Free University of Berlin for helpful comments and suggestions. Fu and Lu gratefully acknowledge the financial support from MOE Singapore (WBS# R313-000-107-112 and WBS# R122-000-155-112, respectively). All errors remain ours.

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