Optimal Prize Structures in Elimination Contests

We consider multi-stage elimination contests where agents' efforts at different stages generate some output for the principal. Depending on the output function various prize structures can be optimal. If the output function depends much more on efforts applied at later stages than on those applied at the earlier ones, the optimal prize structure can be non-monotone, that is, at some stages prizes fall and the agents who are more successful may earn less. Necessary and sufficient conditions for the optimality of such structures are provided. We also show that for any increasing prize shape there exists an output function such that this prize shape is optimal. Further, we consider the case of limited liability, where the principal is not allowed to use negative prizes but can choose a contest success function (CSF). There is always an efficient equilibrium under which the principal is able to extract the full surplus from the agents and the corresponding optimal prize structure is always increasing. Moreover, under some plausible assumptions, the optimal CSF is necessary convex, which corresponds to the most frequently used prize schemes in practice