Tournaments with a Standard

TL;DR

This paper analyzes optimal performance standards and prize schemes in tournaments with rank-dependent rewards, identifying conditions under which specific schemes maximize expected outcomes based on noise distribution properties.

Contribution

It characterizes the optimal performance standard and prize scheme in tournaments, linking them to properties of the noise distribution such as failure rate and likelihood ratio.

Findings

Optimal standard is at a performance mode weakly above the global mode.

Winner-take-all scheme is optimal for increasing failure rate distributions.

Equal prizes are optimal for decreasing failure rate distributions.

Abstract

We study tournaments where winning a rank-dependent prize requires passing a minimum performance standard. We show that, for any prize allocation, the optimal standard is always at a mode of performance that is weakly higher than the global mode and identify a necessary and sufficient condition for it to be at the global mode. When the prize scheme can be designed as well, the winner-take-all prize scheme is optimal for noise distributions with an increasing failure rate; and awarding equal prizes to all qualifying agents is optimal for noise distributions with a decreasing failure rate. For distributions with monotone likelihood ratios -- log-concave and log-convex, respectively -- these pay schemes are also optimal in a larger class of anonymous, monotone contracts that may depend on cardinal performance.