Optimal screening contests

We study the optimal design of contests as screening devices. In an incomplete information environment, contest results reveal information about the quality of the participating agents at the cost of potentially wasteful effort put in by these agents. We are interested in finding contests that maximize the information revealed per unit of expected effort put in by the agents. In a model with linear costs of effort and privately known marginal costs, we find the Bayes-Nash equilibrium strategy for arbitrary prize structures ($1=v_1 \geq v_2 \dots \geq v_n=0$) and show that the equilibrium strategy mapping marginal costs to effort is always a density function. It follows then that the expected effort under the uniform prior on marginal costs is independent of the prize structure. Restricting attention to a simple class of uniform prizes contests (top $k$ agents get $1$ and others get $0$), we find that the optimal screening contest under the uniform prior awards half as many prizes as there are agents. For the power distribution $F(\theta)=\theta^p$ with $p\geq 1$, we conjecture that the number of prizes in the optimal screening contest is decreasing in $p$. In addition, we also show that a uniform prize structure is generally optimal for the standard objectives of maximizing expected effort of an arbitrary agent, most efficient agent and least efficient agent.