Restricting Entries to All-Pay Contests

TL;DR

This paper analyzes an all-pay contest with a filtering mechanism that admits only top players, showing how this affects equilibrium strategies and optimal admission numbers, especially under winner-take-all prizes.

Contribution

It introduces the concept of an inflated ability to capture effects of elimination and characterizes equilibrium strategies in this setting.

Findings

Effort decreases as admitted players increase.

Optimal to admit only two players for maximum effort.

No symmetric equilibrium exists in a two-stage extension.

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.