Creating Balance in Dynamic Competitions
Rivals often meet many times in competitions, sometimes gaining wins and sometimes suffering losses. We consider how the win-loss pattern affects effort incentives over time, and how a principal can divide up a prize fund in order to incite the rivals to exert effort.
The stage game is a two-player all-pay auction, and contestants are heterogeneous in the sense that one of them has a bias in the first contest, meaning that his efforts are more productive than those of the rival. The winner of a contest gains an advantage in future ones, making his efforts more productive; the size of this win advantage also varies between players.
When the initially disadvantaged player is sufficiently able to catch up with the leader, we show how the principal can balance the competition despite the two dimensions of heterogeneity, leading to efforts which exactly match the prize fund in expectation. The optimal reward scheme involves selecting the number of contests to be used, and the prize division between them.