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Although they largely overlap, it is best to include both Mandamus and APA claims in your complaint.
Proving a Mandamus claim requires the plaintiff to show that there are no alternative adequate remedies. Some courts have held that the APA provides an adequate remedy and have dismissed Mandamus claims for this reason. Even so, there really is no clear downside to pleading both, so why run the risk of incurring a technical ruling that could be applied to knock out one but not the other. In economic terms, pleading both provides an asymmetrical risk/reward situation. In other words, you get upside potential with little to no risk, i.e., all the benefits without the costs.