Restricting Entries to All-Pay Contests

TL;DR

The paper analyzes all-pay contests with entry restriction, showing that eliminating entrants can be succinctly captured by an inflated ability that combines true ability with the signaling effect of admission. It proves that a symmetric, strictly increasing Bayesian equilibrium exists if and only if this inflated ability is increasing in true ability, and it derives the explicit equilibrium form. In a winner-take-all setup with two admitted players, the model implies that increasing the admitted number lowers the highest equilibrium effort, making two admissions optimal for maximizing the top effort. Extending to a two-stage sequential elimination contest, the authors show no symmetric and strictly increasing perfect Bayesian equilibrium exists, highlighting the complex interaction between first-stage signaling and second-stage beliefs.

Abstract

We study an all-pay contest in which players with low abilities are filtered out before competing for prizes. We consider a setting where the designer admits a certain number of top players. The admitted players update their beliefs based on the signal that their abilities are among the top, which leads to posterior beliefs that, even under i.i.d. priors, are correlated and depend on each player's private ability. We find that all effects of this elimination mechanism -- including the reduction in the number of admitted players and the resulting updated beliefs -- are captured by an \textit{inflated ability}. A symmetric and strictly increasing equilibrium strategy exists if and only if this inflated ability is increasing in the player's true ability. Under this condition, we explicitly characterize the unique strictly increasing Bayesian equilibrium strategy. Focusing on a winner-take-all prize structure, we find that each admitted player's effort strictly decreases as the admitted number increases. As a result, it is optimal to admit only two players in terms of maximizing the expected highest effort. Finally, in a two-stage extension, we find that there does not exist a symmetric and strictly increasing equilibrium strategy.

Figures (3)

• Figure 1: Comparison between prior and posterior beliefs ($n_1=5$, $n_2=2$ with uniform prior distribution)
• Figure 2: Player $i$'s utility $U_i(\tilde{a}_i,a_i \mid n_2,b)$ ($n_1=10$, $n_2=3$) under a uniform prior and a linear cost function with a winner-take-all prize structure. The vertical axis $U_i(\tilde{a}_i,a_i \mid n_2,b)$ represents the utility of an admitted player $i$ with ability $a_i$ if all other admitted players use the strategy $b(\cdot)$ defined in \ref{['eq_b(a_i)_general_prize']} while she exerts effort $b(\tilde{a}_i)$.
• Figure 3: The jump $H(a_i,n_2\mid F,n_1)$ ($n_1=20$ and uniform prior distribution)

Theorems & Definitions (35)

• Proposition 1: Posterior Beliefs
• proof : Proof Sketch
• Corollary 1: Marginal Posterior Beliefs
• Remark 1
• Example 1
• Proposition 2: Stochastic Dominance of Posterior Belief over Prior Belief
• Lemma 1: Inflation Effect
• Proposition 3
• Corollary 2
• Definition 1: Informal