Two of the most poignant decisions in pediatrics concern disagreements between physicians and families over imperiled newborns. When can the family demand more life-sustaining treatment (LST) than physicians want to provide? When can it properly ask for less? The author looks at these questions from the point of view of decision theory, and first argues that insofar as the family acts in the child’s best interest, its choices cannot be constrained, and that the maximax and minimax strategies are equally in the child’s best interest. He then proposes a guideline according to which the family can demand LST if it is physiologically possible to preserve a life the child can be expected to welcome, and refuse such treatment if it causes suffering that is “more than can be borne” even if an uncompromised life is expected to emerge.