Incentive Effects of a Cut-Off Score: Optimal Contest Design with Transparent Pre-Selection

TL;DR

This paper analyzes rank-order contests with a transparent pre-selection stage that discloses a cut-off and shortlists the top $m$ entrants. It derives a complete symmetric Bayesian Nash equilibrium for admitted contestants under any prize structure and budget, and shows the designer’s optimal prize design is winner-take-all for both HP and TP; specifically, $m^*=2$ for HP and TP’s outcome is invariant to $m$. Through an asymptotic analysis guided by truncated order statistics and beta-function representations, the authors show HP under pre-selection converges to $(2/3)ar{x}$ and yields a $4/3$ gain over no pre-selection, while TP is unaffected by the shortlist. Numerical experiments across several distributions confirm robustness and demonstrate the practical value of the 2-contestant WTA design, with a cautionary finite-population example where larger shortlists can be optimal. The results offer actionable guidance for contest designers in settings with capacity constraints and transparent shortlisting, such as coding contests, crowdsourcing, and hiring processes.

Abstract

Shortlisting is a common and effective method for pre-selecting participants in competitive settings. To ensure fairness, a cut-off score is typically announced, allowing only contestants who exceed it to enter the contest, while others are eliminated. In this paper, we study rank-order contests with shortlisting and cut-off score disclosure. We fully characterize the equilibrium behavior of shortlisted contestants for any given prize structure and shortlist size. We examine two objective functions: the highest individual performance and total performance. For both objectives, the optimal contest is in a winner-take-all format. For the highest individual performance, the optimal shortlist size is exactly two contestants, but, in contrast, for total performance, the shortlist size does not affect the outcome, i.e., any size yields the same total performance. Furthermore, we compare the highest individual performance achieved with and without shortlisting, and show that the former is 4/3 times greater than the latter.

Figures (7)

• Figure 1: Technical Overview for solving the highest individual performance contest design problem. Each box denotes the decision variables of the current optimization problem. Starting from a discrete–high-dimensional continuous program over shortlist size and prize vectors, we progressively simplify the problem through a sequence of technical lemmas into an asymptotic single-variable continuous optimization. This reduction enables a distribution-independent characterization of the optimal contest, yielding a 2-contestant winner-take-all design.
• Figure 2: Highest individual performance under optimal pre-selection across three ability distributions. The optimal contest consistently reduces to a 2-contestant WTA contest and rapidly approaches the asymptotic $4/3$ improvement over the optimal contest without pre-selection.
• Figure 3: Robustness of pre-selection under noisy ability observations ($\tilde{x}_i = (1-\alpha)x_i + \alpha \varepsilon_i$, $\varepsilon_i \sim F$, $\alpha=0.4$). Even when the designer may eliminate high-ability agents due to noise, the induced 2-contestant WTA contest remains close to the ideal benchmark and strictly outperforms the optimal $n$-contestant WTA without pre-selection across all tested distributions.
• Figure 4: Original integration region of $(u,v)$.
• Figure 5: Integration region of $(u,z)$.

Theorems & Definitions (47)

• Definition 1
• Definition 2: Simple Contest
• Definition 3: Winner-Take-All Contest
• Proposition 1: Posterior Beliefs
• Theorem 1: Unique sBNE
• Corollary 1: No Consolation Prize Should be Set
• Proposition 2: The Optimal Contest is a Simple Contest
• Proposition 3: Optimal Prize Structure for the Highest Individual Performance
• proof : Proof Sketch
• Lemma 1: Truncated Distribution