Milgrom and Shannon (1994) provide necessary and sufficient conditions on parameterized optimization problems for their solution sets to be globally monotone in the parameter. We establish that their conditions may be significantly relaxed when focusing on discrete, binary comparisons between solution sets. Such binary comparisons are ubiquitous in economics and may involve comparing the same decision maker across two distinct regimes or two distinct decision makers with related objectives (e.g., a monopolist firm versus a social planner). While the single‐crossing property remains prominent in the theory, quasisupermodularity of the objective functions of interest is not needed. Our approach relies upon a novel method of embedding a new optimization problem with a quasisupermodular objective function “between” the two original problems of interest. In smooth problems, sufficient conditions for our new assumptions may be verified by elementary differential comparisons, making them well suited for applied work. We illustrate the relevance of this novel approach with several economic applications.

Four rankings of economics departments worldwide in terms of graduate education are constructed. The central methodological idea is that the value of a department is the sum of the values of its PhD graduates, as reflected in the values of their current employing departments. Scores are derived as solutions to linear simultaneous equations in the values. The sample includes the top 58 departments, the composition of which is determined endogenously, invoking a criterion requiring more than three placements in the sample. Illuminating the current state and trends of economics PhD education, the conclusions should be of broad interest to PhD candidates, academics, and policymakers.