Optimal grading contests

We study the contest design problem in an incomplete information environment with linear effort costs and power distribution prior $F(\theta)=\theta^p$ on marginal cost of effort. We characterize the symmetric Bayes-Nash equilibrium strategy function for arbitrary prize vectors $v_1 \geq v_2 \dots \geq v_n$ and find that the normalized equilibrium function is always a density function. To study the effects of competition, we compare the effort induced by prize vectors ordered in the majorization order and find that a more competitive prize vector leads to higher expected effort but lower expected minimum effort. We study the implications of these results for the design of grading contests where we assume that the value of a grade is determined by the information it reveals about the quality of the agent, and more precisely, equals its expected productivity. We find that more informative grading schemes induce more competitive prize vectors and hence lead to higher expected effort and lower expected minimum effort.