How to split the pie: Optimal rewards in dynamic multi-battle competitions

Multi-battle competitions are ubiquitous in real life. In this paper, we examine the effort-maximizing reward design in sequentially played multi-battle competitions between two players. The organizer has a fixed prize budget, and rewards players contingent on the number of battles they win in a three-battle contest. A full spectrum of contest technologies in the Tullock family is accommodated. We find that the optimal design varies with the discriminatory power of the contest technology. In particular, when it is in the low range, winner-take-all is optimal. For the intermediate range, as discriminatory power increases, the optimal prize structure evolves continuously from winner-take-all to the proportional-division rule due to the need to mitigate the growing momentum/discouragement effect. For the high range, a wide span of prize structures extracts full surplus and is thus optimal. Several robustness checks confirm that mitigating the momentum/discouragement effect is essential for effort-maximizing prize design in dynamic multi-battle contests.