The algorithm is actually less about matching preferences, even though that is the most common description, than you might think. For example, one of the practical uses is related to matching kidney donors and recipients, andrather than subjective preferences the matching involves sorting out problems related to the compatibility of immune systems. It is more of an engineering problem really. Arindrajit Dube is right that this is a Nobel for planning, if you think about it (h/t Mark Thoma for pointing this out). Also, the stability of the matches (the fact that there is no mutually preferable match) is irrelevant in this context, since who would 'divorce' a kidney that is perfectly compatible from an immunologically point of view (what, you didn't like the kidney's political views?), or which kidney for that matter wouldn't be satisfied with its match?*
It is particularly important that this prize has little to do with the core of neoclassical economics. Yes, the algorithm is about exchanges, and individual decisions, and hence about allocation, and surely has little if anything to say about production (let alone the social relations of production). However, note that it is an idiosyncratic rule, not a general proposition based on the principle of substitution, according to which relative scarcities determine relative prices and individual choices.
This has been a general trend in neoclassical economics since their defeat in the capital debates demonstrated that the latter proposition does not hold water. The use of the disaggregated General Equilibrium model (as I argued here with Kirsten Ford and Nate Cline) has not been able to change the problems from a logical point of view, and that's why they still use the aggregative model to give policy advice (i.e. structural reforms in Europe to cope with unemployment, meaning lower wages so firms hire the cheap 'factor of production').
It is also why prizes for game theory based research are popular with the Swedish bankers that make decisions about who is an authority in this profession. Game theory is an instrument that fits well the post-modern version of the mainstream, in which everything is possible (but the 'free' markets are still the mantra). So beyond the question of what practical use this particular algorithm might have, there is a deeper question of the bankruptcy of the mainstream, that pretends to be more general and flexible, when is exactly the opposite. It is internally illogical and incoherent, but increasingly attached to the dogmatism of 'free' markets, which has become just a way to defend the interests of the wealthy and corportaions (any similarity to the Conservative movement, and the current GOP in the US, is NOT a coincidence).**
* I'm not suggesting that the algorithm works perfectly in the real world. In fact, in Brazil the National Association of Graduate Economic Centers (ANPEC, in Portuguese), following Marilda's lead, adopted the system in 1997 to match students with centers to produce a stable matching, that is, there would be no mutually preferable matches. Note that before good students were sometimes left without a graduate center, since they were not chosen by anyone (even though some students that had lower grades did), and a few centers were left with empty vacancies. However, as noted in this dissertation supervised by Marilda (in Portuguese), the algorithm did NOT work, and a stable matching was not achieved, with the same problems taking place as before. The algorithm was abandoned the following year.
** Supply-side economics, New Classical and Real Business Cycles authors (as I claimed in another post) are the equivalent in economics to Intelligent Design in Biology, and the GOP has adopted all of these views as part of their world view.