Optimal rank-order tournaments have traditionally been studied using a first-order approach. The present analysis relies instead on the construction of an "upper envelope" over all incentive compatibility conditions. It turns out that the first-order approach is not innocuous. For example, in contrast to the traditional understanding, tournaments may be dominated by piece rates even if workers are risk-neutral. The paper also offers a strikingly simple characterization of the optimal tournament for quadratic costs and CARA utility, as well as an extension to large tournaments.
Ewerhart, C. (2016). An envelope approach to tournament design. Journal of Mathematical Economics, 63, 1–9. https://doi.org/10.1016/j.jmateco.2015.11.004