  Probabilistic record linkage is often used to match records from two files,
in particular when the variables common to both files comprise imperfectly
measured identifiers like names and demographic variables. We consider
bipartite record linkage settings in which each entity appears at most once
within a file, i.e., there are no duplicates within the files, but some
entities appear in both files. In this setting, the analyst desires a point
estimate of the linkage structure that matches each record to at most one
record from the other file. We propose an approach for obtaining this point
estimate by maximizing the expected $F$-score for the linkage structure. We
target the approach for record linkage methods that produce either (an
approximate) posterior distribution of the unknown linkage structure or
probabilities of matches for record pairs. Using simulations and applications
with genuine data, we illustrate that the $F$-score estimators can lead to
sensible estimates of the linkage structure.
