Optimal Matching with Matching Priority

Matching algorithms are commonly used to build comparable subsets (matchings) in observational studies. When a complete matching is not possible, some units must necessarily be excluded from the final matching. This may bias the final estimates comparing the two populations, and thus it is important to reduce the number of drops to avoid unsatisfactory results. Greedy matching algorithms may not reach the maximum matching size, thus dropping more units than necessary. Optimal matching algorithms do ensure a maximum matching size, but they implicitly assume that all units have the same matching priority. In this paper, we propose a matching strategy which is order optimal in the sense that it finds a maximum matching size which is consistent with a given matching priority. The strategy is based on an order-optimal matching algorithm originally proposed in connection with assignment problems by D. Gale. When a matching priority is given, the algorithm ensures that the discarded units have the lowest possible matching priority. We discuss the algorithm’s complexity and its relation with classic optimal matching. We illustrate its use with a problem in a case study concerning a comparison of female and male executives and a simulation.