We study a sequential all-pay auction where heterogeneous contestants are privately informed about a parameter (ability) that affects their cost of effort. In the case of two contestants, contestant 1 (the first mover) makes an effort in the first period, while contestant 2 (the second mover) observes the effort of contestant 1 and then makes 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. This model is then generalized to any number of contestants where in each period of the contest, 1 /< j /< n, a new contestant joins and chooses an effort. Contestant j observes the efforts of all contestants in the previous periods and then makes an effort in period j. He wins if his effort is larger than or equal to the efforts of all the contestants in the previous periods and strictly larger than the efforts of all the contestants in the following periods. This generalized model is studied also with a "stopping rule" according to which the contest ends as soon as a contestant exerts an effort strictly smaller than the effort of the previous contestant. We characterize the unique sub-game perfect equilibrium of these sequential all-pay auctions and analyze the use of head starts to improve the contestants' performances.
