Less is More: Optimal Contest Design with a Shortlist
摘要
This paper examines the optimal design of contests with n contestants, where the designer determines both the number of admitted contestants in a shortlist and the prize structure under a fixed budget. For the maximum individual effort objective, the optimal contest is remarkably simple—a two-contestant winner-take-all contest, where only two contestants are shortlisted, and a single prize is awarded. For the total effort objective, the optimal contest admits \(m=kn\) contestants and distributes the budget equally among the top \(m-1\) contestants. We also present a general algorithm to determine the optimal k for any ability distribution (e.g., \(k=0.09\) for exponential distribution and \(k=0.15\) for the uniform distribution), which has practical applications. Additionally, we establish a tight upper bound of \(\bar{k}=0.3162\) for any distribution. Finally, we compare total effort across different contest configurations: the optimal contest without a shortlist achieves \(\varTheta (1)\) , a two-contestant winner-take-all contest yields \(\varTheta (\log n)\) , and the optimal contest with a shortlist reaches \(\varTheta (n)\) . This demonstrates the significant advantage of incorporating a shortlist.