Beyond Gale-Shapley, or, Analysis As Fad
I was at a party recently, and some people were trying to discuss some known-to-the-ingroup person who was saying, hey, Gale-Shapley, women should ask out men!
I do not blame them for discussing this per se. In the folly of my youth, I said this precise thing. I suspect my comments filtered to the person in question, in the way that happens in social circles, and he added some insignificant additional analysis and spread it on twitter, and his whole life consists of this type of barely-thoughtful gibberish, and he's actually totally vapid. Or someone made a youtube video a year ago or whatever. These things sort of grow on their own. If someone can show me this comment before 2009 I'll concede my younger self fits the pattern (although without the megaphone) -- I was quite a moron, but would have been so impressed with myself for the analysis it explains why I recall mentioning this to so many people at the time.
Today, this has become perhaps my most troublesome pet peeve, as only our own faults can be. I desperately wanted the conversation to avoid this topic, for I could hear how awfully vapid it was, and eventually got my way by luring people into discussing whether self-selecting media bias means voting is Monty-Hall-like and you should take a preference (like for Libertarianism), get news about one of the other options being bad (from a major party), and update towards "switching doors" to the other major party candidate. A sweet release from my personal torture.
The reason the Gale-Shapley-means-women-should-ask-out-men thing bothers me so much is hard to explain, but let me try: I would be willing to bet all the money in my pocket not a single person there but me and an anonymous thoughtful person (who was who I targeted for the Monty Hall Politics diversion) could describe Hall's condition for how to notice a bipartite graph is deficient for this matching problem. It's a foundational result! It's extremely simple! You could explain it in conversation quickly and simply assert it's truth, although the proof is pretty short -- admittedly more technical than most have the patience for.
Not a single person there, I'd bet, could explain the Gallai-Edmonds decomposition of bipartite graphs. And therefore they'd be ill-equipped to understand Bogomolnaia's model involving dichotomous preferences, which solves a major gap in the model used by Gale-Shapley, notably, everyone has partners for which the prospect of marriage is less appealing than being single. But there are of course many gaps in these models. For instance, if you've not been in a relationship where you begin it thinking, wow, they're really out of my league, but eventually realize, oh, you're perfect except you don't like me nearly as much as I like you... then you don't realize you aren't ranking or preferring partners, but relationships. But then preference falsification seems inevitably strategic -- you want to find someone who would want you even if you don't want them, so that if you do want them you're ensured a very good outcome, perhaps. That kind of strategic preference falsification isn't the only type, but plausible in some models we might imagine. My whole point here is that these models are extremely simple, to the point that they miss the actual dynamics in play. This is good! A sign of a good model! You want it to be simple enough to get a result. But you can't talk about about "this model implies this algorithm implies these results implies if you want different results you have to change the model this one specific tiny way...", you're simply not engaging in the same project as these researchers, and you're sort of dragging their names through the mud. I'm not even entirely sure they could understand, once described, the lattice of stable matchings even though that's actually a critical element of their reasoning -- if you would call it that. It's not their reasoning, that's what bothers me so much. No point in even discussing it.
It's fine if people don't want to talk about these actual results, and want to use it as a jumping off point for a broader discussion. But if mentioning the details of the papers is out of bounds, please don't use them to justify your analysis. I have no real reason to dislike it as much as I do, but it bothers me more than a rock in my shoe.