The Tragic Flute

I found this little thought experiment, inserted by a Forbes editor into an interview with Amartya Sen, pretty peculiar.

[Note: In the book, Sen describes a problem of divergent views on justice in which you have one flute and three children who want it. One child wants the flute because she knows how to play it, the second one wants it because he is poor and doesn't have toys, and the third one says she made the flute, so she should get it. Who do you give it to?]

This is no knock against Sen, since there's probably more context in the book. But this is not really such a puzzling question, is it? The correct answer is: It all depends on how “you” ended up with the flute!
Is the flute yours because you provided the materials (which were yours) and paid the kid who made it? If so, you can give it to anyone you want, or you can keep it. It's yours! Did you steal it from the kid who made it? Then you should give it to the kid who made it. It's hers! You've got no right to redistribute her flute.
Anyway, I find this thought experiment, and the not uncommon practice of assuming away the relevance of property rights when considering questions of distributive justice, confusing. A settled scheme of property rights is the main solution to the problem of distributive justice. As Hume said:

No one can doubt, that the convention for the distinction of property, and for the stability of possession, is of all circumstances the most necessary to the establishment of human society, and that after the agreement for the fixing and observing of this rule, there remains little or nothing to be done towards settling a perfect harmony and concord.

Of course, ingrained intuitions about equality, reciprocity, need, desert etc. will tend to shape this most necessary convention as it develops over time. One of the things we already know before coming to questions about who should get a flute is that things like flutes don't grow on trees. Somebody has to make them. And the production of valuable goods, like flutes, tends to take place within a stable scheme of property. It's sort of silly to drop that context and ask who “you” should give the flute to. If the conditions under which flutes are produced is in already place, then the answer about who should get it is likely already settled.
Anyway, Sen's point no doubt was just that if you're trying to do the best you can in giving away your rightful possessions, considerations of desert (e.g., both “I made it” and “I can best use it”), need, and fairness (e.g., “I don't have ANY toys!”) are definitely relevant, but it's not clear how to weigh them against one another.
What I'd add is that because the relative importance of considerations such as desert, equality, need, reciprocity, etc. tends to be indeterminate in the abstract, we need a system of property to peacefully settle particular distributional questions. Stable property conventions will tend to have evolved over time in a way that has taken the various elements of justice into account. If they didn't, they'd likely lead to conflict and wouldn't be stable. But we shouldn't be surprised that the aggregate pattern of holdings that emerges from cooperation and exchange according to a just scheme of property rights will fail to resemble a counterfactual pattern determined solely by considerations of desert, or of equality, or of need.
Update: Also, I like what Sen says at the end of the interview:

The idea of perfect justice drives to a mistaken route. If you believe that any judgment has to be examined through public agreement, reasoned agreement, it's my submission that
a) We won't have agreement on the nature of a perfectly just society. But very likely, we will have agreement, reasoned agreement, on a variety of arrangements, outcomes, social states which are unjust and should be removed.
b) Secondly, even if we succeeded in identifying a perfectly just society on which everybody agrees, that's not what we are debating about. No one expects we can have a perfectly just society in the foreseeable future. Our policies don't depend on it. So why waste your time on a question which is probably unanswerable and certainly redundant.