Optimal prize structures in elimination contests

We consider multi-stage elimination contests where agents’ efforts at different stages generate some output for the principal. 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.
• For any increasing prize shape there exists an output function such that this prize shape is optimal. This is not true for decreasing prize shapes.
• If the principal is not allowed to use negative prizes but can choose a contest success function (CSF) the corresponding optimal prize structure is always increasing. Under some plausible assumptions, the optimal prize scheme is necessary convex, which corresponds to the most frequently used prize schemes in practice.