Artificial Intelligence has the potential to improve human decisions in complex environments, but its effectiveness can remain limited if humans hold context-specific private information. Using the empirical example of antibiotic prescribing for urinary tract infections, we show that full automation of prescribing fails to improve on physician decisions. Instead, optimally delegating a share of decisions to physicians, where they possess private diagnostic information, effectively utilizes the complementarity between algorithmic and human decisions. Combining physician and algorithmic decisions can achieve a reduction in inefficient overprescribing of antibiotics by 20.3 percent.