Optimal Contest Design with Entry Restriction

TL;DR

This paper studies how to design a rank-order contest under entry restrictions to maximize contestants' effort, with the designer choosing the shortlist size $m$ and prize structure $\vec{V}$ within budget $B$. It derives a unique symmetric Bayesian Nash equilibrium for admitted contestants, characterizes posterior beliefs after admission, and uses this to solve the optimal contest design for two objectives: maximum individual effort and total effort. For maximum individual effort, the optimal design is a two-contestant winner-take-all contest; for total effort, the optimal design is a complete simple contest with the shortlist size growing linearly in $n$ and an asymptotic upper bound of $m^*/n \le 0.3162$, with the optimal prize scheme having equal nonzero prizes and one zero-prize. Across distributions, shortlisting yields substantial improvements over no-shortlist designs, achieving up to $\Theta(n)$ growth in total effort and a $\Theta(\log n)$ gain in maximum individual effort, and the paper provides practical algorithms and guidelines for implementation.

Abstract

This paper explores the design of contests involving $n$ contestants, focusing on how the designer decides on the number of contestants allowed and the prize structure with a fixed budget. We characterize the unique symmetric Bayesian Nash equilibrium of contestants and find the optimal contests design for the maximum individual effort objective and the total effort objective.

Figures (6)

• Figure 1: Posterior beliefs of player $x_1$ ($n = 5,m=2,F(x)=x^2$).
• Figure 2: The actual optimal size and $m^*$ predicted by asymptotic relation.
• Figure 3: Supremum of optimal $m$.
• Figure 4: A flow chart for contest design in practice.
• Figure 5: Total effort performance of different contest designs.

Theorems & Definitions (98)

• Theorem 1: Unique sBNE of Contestants (Sketch)
• Theorem 2: Optimal Contest for the Maximum Individual Effort Objective
• Theorem 3: Optimal Contest for the Total Effort Objective
• Theorem 4: Tight Upper Bound for the Optimal Shortlist Size
• Theorem 5
• Theorem 6
• Definition 1
• Proposition 1: Posterior Beliefs
• Corollary 1: Marginal Posterior Beliefs
• Proposition 2: Stochastic Dominance of Posterior over Prior