Competitive balance when winning breeds winners

In contest settings, heterogeneity between contestants generally leads to low effort provision, and many instruments have been suggested to restore competitive balance. We suggest that heterogeneity may evolve depending upon the outcome of previous contests. Restoring competitive balance in this setting is challenging, and we consider the choice faced by a principal who can distribute a prize fund over two consecutive all-pay auction contests. Contestants are heterogeneous in the sense that one of them has a head start in the first contest. The winner of contest one gains the stage prize and a net head start in contest two; the size of this net head start also varies between players. The head start of the early leader can be increased, neutralized or overturned. The principal aims to maximize total effort. We show how the potential head start that the initial underdog can gain is critical in determining expected efforts. When this parameter is small, the principal can do no better than running a single contest. When the initial underdog can catch up and surpass the first-contest favourite, there is scope to do better. Indeed, it is possible to completely neutralize the initial heterogeneity and capture all of the surplus from the players, inciting a level of expected effort equal to the prize value. Competitive balance is achieved by a judicious division of the prize budget.