Optimal Grading: A Unified Approach

Bin Liu and Jingfeng Lu

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Abstract: We develop a unified approach to optimal grading in an all-pay contest in which a designer assigns a fixed vector of heterogeneous prizes to maximize expected total effort. The approach covers two information regimes and identifies a common principle: iron locally misordered incentive returns and assign prizes assortatively across the resulting grades. Under rank-only grading, assignments depend only on ordinal ranks. Ironing cumulative rank coefficients---via the least concave majorant or the pool-adjacent-violators algorithm---determines which adjacent ranks are pooled and which prizes are randomized within each grade. Under performance-contingent grading, assignments may depend on numerical effort. The optimum irons virtual ability, forms endogenous type grades, and assigns prize blocks assortatively across grades. A failing grade below a minimum passing effort and a collection of effort brackets implement the direct optimum while preserving full prize assignment.

Date: 2026-08
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