Thanks for taking the time to reply in depth. I think this is a situation with a bit of nuance, and it's possible to really endlessly argue about the issue if we don't tie down the definitions a bit.

> You should also consider that if we got each others’ best preference, this is a sign of a broken system, because we could have clearly been assigned to our first choices in the first place. So in my view, the exchange should be allowed, and then the system fixed so that such situations are not possible.

The thing is that matching systems have been studied for a long time and there is a very rich game theoretic literature that studies this both from a pure theory perspective (where we assume people's preferences are given), and also in real life (e.g. the doctor matching system in the US).

What you're demanding here is that the matching system is finds a stable matching, such that there are no trades that any pair of players would be willing to make.

This is an actual goal of most matching systems. The problem is that it conflicts with the goal of relieving participants from making strategizing choices[1] [2].

Not to get into the weeds to much, but you can see that in the classical stable marriage problem, where only men propose and women either accept or reject, the matching is only stable amongst the men: no man would be willing to switch his partner with another man, but there will (in general) be women who are willing to switch her partner with another woman.

The male-proposed stable matching is actually locally man-optimal and woman-pessimal, in the sense that any trade that a pair of women decide on would by definition make their current fiances worse off.

> How? They will be in the same school regardless of other students switching or not. And even if you didn’t strategize, you might receive an offer for an exchange, so you could even end up just as well as someone who did strategize.

In the sense that relative wellbeing matters to people in general. A system that provides half of people $1 and everyone else $10 makes people better in absolute terms, but worse off in relative terms, and that matters to people because access to many desirable goods in society depend on people's relative ranking.

This is a very well-understood effect and has been measured in different contexts, e.g. [3]

You are right that a person that does no strategizing may still receive an exchange offer, but it should be easy to show that in general, effective strategizing increases the probability of an exchange offer and the net quality of the school.

> I don’t get where this idea that globally the students will be better off if exchanging places is forbidden. Some people will be better off by exchanging, others will be in the same position as they were before. It’s clear that globally the system will have gained from more people being at their preferred schools.

I agree with you that exchanges increase welfare locally, and it would be nice to come up with a system that by definition come up with matches that are inherently stable to any pairwise trades, but as far as I'm aware there is no matching system that simultaneously does that and also effectively discourages strategic behavior (e.g. 'lying' about preferences). Game theory is full of proofs of the impossibility of simultaneously reaching 'obviously' desirable goals, e.g. [4].

Note that an equilibrium where consumers are honest about their preferences and then the 'matchmaker' can rely on those preferences is much simpler evaluate and optimize. This is why, for example, auctions are an effective way to find price—bids are generally close to 'honest' reflections of reservation prices.

As soon as broad strategic behavior comes into play, it's becomes much harder to determine whether your system is actually working well since the preferences no longer reflect what people really want.

A system where everyone is essentially masking their true preferences imposes obvious costs: everyone who doesn't strategize will be worse off relatively than those who do. Strategizing costs people time and energy. It also makes the matchmaker's job obviously harder, since the 'first step' would be to try to back out people's true preferences.

We can certainly argue about whether those costs are higher or lower than the cost of not sitting at a local optimum, but it seems unwise to simply deny that there is any tradeoff, or to completely ignore the fact that people care deeply about relative welfare as well as absolute welfare.

[1] https://arxiv.org/pdf/1511.00452.pdf

[2] https://arxiv.org/abs/1608.07575

[3] https://en.wikipedia.org/wiki/Ultimatum_game#Explanations

[4] https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theore...