The eponymous dilemma of the play is that of the newly knighted doctor Sir Colenso Ridgeon, who has developed a revolutionary new cure for tuberculosis. However, his private medical practice, having limited staff and resources, can only treat ten patients at a time. From a group of fifty patients he has selected ten he believes he can cure and who, he believes, are most worthy of being saved. However, when he is approached by a young woman, Jennifer Dubedat, with a deathly ill husband, Louis Dubedat, he admits that he can, at a stretch, save one more patient, but he insists that the individual in question must be shown to be most worthy of being saved. The situation then becomes complicated, as an old friend and colleague reveals, he too, needs treatment. Sir Colenso must choose which patient he will save: a kindly, altruistic, poor medical colleague, or a young artist who is extremely gifted, but an amoral, very unpleasant womaniser and bigamist. Sir Colenso has also fallen in love with the young and vivacious Jennifer Dubedat, which makes it even harder for him to be clear about his motives for choosing who shall live.
In the end, the doctor decides to treat his colleague, deceiving Louis and Jennifer by telling them that another doctor will treat him. Before Louis dies, he persuades his wife to promise that she will remarry. Sir Colenso convinces himself that she will marry him. But when she learns of his wish, she is horrified: she cannot imagine wanting to marrying someone as old as he is. At that point, Sir Colenso confesses to her that he had selected his colleague over Louis partly because he is infatuated with her. Jennifer’s disgust upon learning this makes Sir Colenso question his decision.
