Great Math Quiz

Problem found first at The Volokh Conspiracy.  Actually it's pretty obvious.

Another, more serious puzzler:

Al’s favorite hobby is the piano. One day he has a traffic accident. As he is pulled from the rubble he notices that the dexterity of one of his fingers is somehow impaired. He is greatly worried, and goes to an emergency room, where he finds exactly one physician in attendance. It quickly turns out that time here is of the essence, if Al’s finger is to be restored to its former piano-playing level. Luckily there is no other patient competing for the doctor’s attention.

But as the doctor is about to treat Al, something unfortunate happens. For just then Bea is brought in, who has suffered a serious injury to one of her legs, which will turn into paralysis unless immediately taken care of. Loss of one’s leg being a bigger problem than a slight loss in digital dexterity, the doctor of course immediately turns from Al to Bea.

And then something else very strange happens. Al notices that he has suffered another injury he and the doctor have overlooked so far—an injury which, if not taken care of immediately, will cause him to lose the use of both his legs. By virtue of this fact, he of course immediately regains the doctor’s attention.

Then things get even stranger. For Al asks the doctor to please ignore his legs for the time being and just focus on making sure his finger is fully restored. At first the doctor refuses. “How can I possibly justify not dealing with Bea’s more serious leg injury for the sake of dealing with your overly intense concern about your finger.” “Never mind,” says Al, “Bea is not going to be treated, no matter what. If you don’t treat my finger, you will have to treat my legs. So Bea is truly out of the running. She is not going to get priority. And given that you are committed to treating me, wouldn’t you do better if you did what I preferred, which is to have my finger rather than my legs treated first? Isn’t that a win-win transaction?”

The doctor now has to choose among three alternatives: Treating Al’s hands first, treating Bea’s leg first, or treating Al’s two leg injury first. Each of these seems for different reasons unacceptable.

How can he treat Al’s finger when Bea’s more serious leg injury needs attention? How can he treat Bea’s leg injury when Al’s far more serious leg injury isn’t getting treated? How can he treat Al’s leg injury when Al clamors to have his finger treated instead? We are in a sort of cycle.

How is it to be broken? At least one plausible way of breaking it, my own guess is that it is the most plausible, is to not let Al opt for getting his finger treated. He retains his priority over Bea only if he has is leg injury treated, not otherwise. But that means that we are rejecting a win-win transaction. For that is what the doctor’s treating Al’s finger rather than his legs surely is.
But why exactly is that?

Why exactly shouldn’t the doctor accommodate Al’s preferences? The reason, in brief, is that Al has a strong claim on the doctor’s attention for his serious injury and a much weaker claim for his less serious injury. That’s a common phenomenon—having a strong claim to something we care less about as compared to something else we care more about but have a less strong claim to. (We may value what belongs to someone else more than we value something that belongs to us, to pick the most rudimentary illustration.)

If the doctor were to attend to what Al has a strong preference but a weak claim to—treat his finger—Bea would rightly be able to protest: “What, for the sake of a mere finger, I am to lose the use of my leg.”

 Learn to play the piano without the finger, Al.