Optimal Contest Design with Incomplete Information ∗ Bin Liu † Jingfeng Lu ‡ Ruqu Wang § Jun Zhang ¶ January 13, 2015 Abstract In this paper, we characterize the optimal contest mechanism with independent private abilities of the contestants. The contest designer has a fixed prize budget to extract effort from the contestants, and both positive and negative prizes are allowed. We find that there exists no optimal contest mechanism maximizing the total effort from the contestants. Nevertheless, by invoking exploding negative prizes, the designer can extract effort approaching the utmost level (i.e., highest possible effort inducible when all contestants are of the maximum ability with certainty). When a bound (i.e., K) on the negative prizes is imposed, we fully characterize the optimal contest mechanism, which can be implemented by a modified all-pay auction with a minimum bid and an entry fee of K per contestant. JEL Classification Numbers: D72, D82, C73 Keywords: optimal contest, mechanism design, negative prize, incomplete information, cross-type transfer, leverage * We are grateful to James Atsu Amegashie, Yaron Azrieli, Tilman B¨ orgers, Yi-Chun Chen, Soo Hong Chew, Jeff Ely, Drew Fudenberg, Robert Hammond, Paul Healy, Johannes H¨ orner, Yuan Ju, Jinwoo Kim, Peter Klibanoff, Charles Knoeber, Fuhito Kojima, Georgia Kosmopoulou, Takashi Kunimoto, Roger Lagunoff, Stephan Lauermann, Dan Levin, Xiao Luo, George Mailath, Thomas Mariotti, Preston McAfee, Diego Moreno, Thayer Morrill, Wojciech Olszewski, Alessandro Pavan, James Peck, Bob Roberts, Ariel Rubinstein, Yuval Salant, Mark Satterthwaite, Klaus Schmidt, Paul Schweinzer, Xianwen Shi, Ron Siegel, Ning Sun, Yeneng Sun, Satoru Takahashi, Flavio Toxvaerd, Rakesh Vorah, Quan Wen, Asher Wolinsky, Huanxing Yang, Zaifu Yang, Lixin Ye, Xiaoyong Zheng and participants at various conferences for very helpful comments and suggestions. Lu gratefully acknowledges the financial support from MOE of Singapore. Wang gratefully acknowledges the financial support from SSHRC of Canada. Zhang thanks University of Technology Sydney for financial support under Business Research Grant. The usual disclaimer applies. † Bin Liu: Department of Economics, National University of Singapore, Singapore 117570. Email: binliu29@u.nus.edu. ‡ Jingfeng Lu: Department of Economics, National University of Singapore, Singapore 117570. Email: ecsljf@nus.edu.sg. § Ruqu Wang: Department of Economics, Queen’s University, Ontario K7L 3N6, Canada. Email: wangr@queensu.ca. ¶ Jun Zhang: Economics Discipline Group, School of Business, University of Technology Sydney; Tel: +61 2 95143203, Fax: +61 2 95147722, E-mail: zhangjunqueens@gmail.com.