Beating the Matthew effect: Head starts and catching up in a dynamic all-pay auction

We consider an effort-maximizing principal distributing a prize fund over two consecutive all-pay auctions. The two contestants are doubly heterogeneous: one of them has a head start in the first contest; and winning contest one gives an advantage in contest two that varies between players. We show that, with a large head start, the principal chooses a zero prize in contest two, i.e., runs a single contest. Otherwise, the laggard winning contest one may overturn the leader 's head start, possibly inciting expected efforts equal to the prize value, avoiding the laggard giving up, and this way mitigating the Matthew effect.