Sequential All-Pay Auctions with Head Starts and Noisy Outputs

We study a sequential (Stackelberg) all-pay auction with two contestants who are privately informed about a parameter (ability) that affects their cost of effort. Contestant 1 (the fi?rst mover) exerts an effort in the fi?rst period, while contestant 2 (the second mover) observes the effort of contestant 1 and then exerts an effort in the second period. Contestant 2 wins the contest if his effort is larger than or equal to the effort of contestant 1; otherwise, contestant 1 wins. We characterize the unique subgame perfect equilibrium of this sequential all-pay auction and analyze the use of head starts to improve the contestants' performances. We also study this model when contestant 1 exerts an effort in the fi?rst period which translates into an observable output but with some noise. We study two variations of this model where contestant 1 either knows or does not know the realization of the noise before she chooses her effort. Contestant 2 does not know the realization of the noise in both variations. For both variations, we characterize the subgame perfect equilibrium and investigate the effect of a random noise on the contestants' performance.