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.