Judges with complex or contentious cases on their dockets have a larger and more diverse list of lawyers to appoint as special masters—just don’t call them special masters.

Special masters, appointed by a judge and paid for by the parties, are brought in on matters that overwhelm the court, such as intense discovery disputes or communications with thousands of class members. But for years, the same individuals have dominated the profession—most of them older, white men from Big Law firms.