Yes. Given the set-up, it's a standard prisoners' dilemma (well, at least if you add the proviso that it's guaranteed to be one-shot). Iff you can ensure that the other doctor will co-operate iff you co-operate, then co-operate. If not defect.

(Or, more accurately, a probabilistic version of this. E.g. assuming you are risk neutral, co-operate iff the other doctor has at least a 50% greater chance of co-operating if you co-operate than if you defect.)