How to Split the Pie: Optimal Rewards in Dynamic Multi-Battle Competitions

Abstract

Multi-battle competitions are ubiquitous in real life. In this paper, we examine the effort-maximizing reward design in sequentially played multi-battle competitions. The organizer has a fixed prize budget, and rewards players contingent on the number of battles they win in a three-battle contest. Battles are played between two opposing players or between selected pairs of players from two opposing teams. A full spectrum of contest technologies in the Tullock family is accommodated. We find that the optimal design is implemented by a contest prize for the grand winner who wins the majority of battles together with uniform battle prizes to battle winners. For competitions between two individuals, the optimal design varies with the discriminatory power of the contest technology. 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 a proportional-division rule due to the need to mitigate the growing momentum/discouragement effect. For the high range, a whole span of prize structures extracts full surplus and thus is optimal. In contrast, winner-take-all is optimal for team competitions, regardless of the contest technology, in which the momentum/discouragement effect does not exist.

Keywords: Effort Maximization, Multi-Battle Contest, Proportional-Division Rule, Split-Award, Team Contest with Pairwise Battles, Winner-Take-All

JEL Classification: D72, D74, D81