Optimal elimination contest

We consider multi-stage elimination contests, where agents efforts at different stages generate some output for the organizers. Depending on the output function we characterize the optimal prize structure of the tournament and show that it is almost efficient. We have found that in some cases quite a strange structure turns out to be optimal, under which prizes for agents are smaller at the later stages than at the earlier ones. Sufficient conditions for optimality of such structures are provided for the case of a separable output function. Next we consider the modi cation, when the designer can specify a winning function. We provide sufficient conditions for optimality of a winning function and show that it can be found in the class of Tullock functions. This function does not depend on the output function. There is always an efficient equilibrium, under which the designer is able to extract the whole surplus from the agents and the corresponding optimal prize structure is always non-decreasing.