🤖 AI Summary
This paper investigates how rank-based admission screening—admitting only the top-k contestants—in all-pay contests affects Bayesian equilibrium and effort incentives. Using a Bayesian game model, we characterize players’ posterior ability distributions formed through belief updating under the elimination rule, and identify, for the first time, an equivalence between this mechanism and a “capability inflation” effect. We rigorously establish necessary and sufficient conditions for the existence of a symmetric, strictly increasing Bayesian equilibrium, fully characterizing its unique solution. We show that increasing k systematically reduces individual effort, and that k = 2 maximizes the highest equilibrium effort. Extending to a two-stage contest, we prove that no symmetric, strictly increasing equilibrium exists. Our core contribution lies in endogenizing the screening mechanism as a belief-driven reconfiguration of competitive dynamics, and providing testable conditions for equilibrium existence and optimal contest design.
📝 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 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.
