Resource Allocations In Multi Stage Contests

We study best-of-k contests (k=2,3) between two players. The players have heterogeneous resource budgets that decrease within the stages proportionally to the resource allocated in the previous stages such that for each resource unit that a player allocates, he loses [alpha] (the fatigue parameter) units of resources from his budget. We show that in both contest forms, independent of the values of the fatigue parameters, each player allocates his smallest resource in the last stage. In the best-of three contest where there are different fatigue parameters for each of the two first stages, a sufficient condition that the resource allocation in the first stage is larger than in the second one is that the value of the fatigue parameter of the first stage is smaller than or equal to the value of the fatigue parameter of the second stage. We also show that in the best-of-three contest, if the fatigue parameters are sufficiently large (approaches one), both players allocate almost all their resource budgets in the first two stages such that they have no resources left for the last stage in which the winner might be decided.