Effort allocations in elimination tournaments

We study elimination tournaments with n stages and 2ⁿ symmetric players. The players have effort budgets that decrease within the stages proportionally to the efforts allocated in the previous stages such that for each effort unit that a player allocates, he loses α (the fatigue parameter) units of effort from his budget. We show that if the fatigue parameter α is larger than [Formula presented], the players allocate their efforts equally over all the first t stages, t≤n−1. In particular, if the fatigue parameter α is larger than [Formula presented], the players allocate their efforts equally over all the first n−1 stages, and only in the final stage, they exert a lower effort.