Asymmetric all-pay contests with heterogeneous prizes

This paper studies complete-information, all-pay contests with asymmetric players competing for heterogeneous prizes. In these contests, each player chooses a performance level or “score”. The first prize is awarded to the player with the highest score, the second – less valuable – prize to the player with the second highest score, etc. The players are asymmetric as they incur different scoring costs, and they are assumed to have ordered marginal costs. The prize sequence is assumed to be either geometric or quadratic. We show that each such contest has a unique Nash equilibrium, and we exhibit an algorithm that constructs the equilibrium. Then, we apply the results to study the issue of tracking in schools and the optimality of winner-take-all contests.