While the zero-sum nature is unavoidable
I believe this is false as stated:
While 'zero-sum' is correct in a loose colloquial sense that at least one person has to lose something for any group to improve, I think it's actually important to realize that there are mechanisms that improve overall welfare -- and so the system administrators should be trying to find them!
No, I think it's a classic scarce resource allocation, and among the parents it's a zero-sum game in the stronger technical sense too. It's the metagame the administrators play of choosing the game the parents play where it's possible to do better, where the "zero" the parents compete around can be increased.
With respect, I think that's wrong.
If all parents agree that school A is better than B, but parent 1 cares much more about A>B than parent 2 does, then the sum-of-utilities is different (so, not "zero sum") depending on whether [ 1→A; 2→B ] or [ 1→B; 2→A ]. Every change in outcomes leads to someone losing (compared to the counterfactual), but the payoffs aren't zero-sum.
That example is kind of useless, but if you have three parents and three schools (and even if parents agree on order), but each of the parents care about A>B and B>C in different ratios, then you can use that fact to engineer a lottery where all three parents are better off than if you assigned them to schools uniformly at random. (Sketch of construction: Start with equal probabilities, and let parents trade some percentage of "upgrade C to B" for some (different?) percentage of "upgrade B to A" with each other. If they have different ratios of their A>B and B>C preferences, positive-sum trades exist.)
Then, in theory, a set of parents cooperating could implement this lottery on their own and agree to apply just to their lottery-assigned school, and if they don't defect in the prisoners' dilemma then they all benefit. Not zero-sum.
Of course, it can also be the case that they value different schools different amounts and a bad mechanism can lead to an inefficient allocation (where pairs would just be better off switching), and I could construct such an example if this margin weren't too narrow to contain it.
It is separately the case that if the administrators have meta-preferences over what parents' preferences get satisfied, then they can make a choice of mechanisms ("play the metagame", as you put it) that give better / worse / differently-distributed results with respect to their meta-preferences.
You cite the Gale-Shapley papers, but are you aware that the school-choice mechanism you described is called the "Boston mechanism" in the field of mechanism design? Because, well, it was also the system in place at Boston Public Schools (until the early 2000s, when they changed to a Gale-Shapley algorithm).
Pathak and Sonmez (2008) is the usual citation on the topic, and they find (as you suggest) that the change makes the most "sophisticated" parent-players worse off, but the least-sophisticated better off.
Interesting! That Boston Public Schools switched from this mechanism to Gale-Shapley seems like it might be useful in convincing our school board (which is separate from the BPS school board, since schools are municipality-level here) to switch.
It might!
In case it would also help to have two-to-three Harvard and/or MIT professors who work on exactly this topic to write supporting letters or talk with your school board, I'll bet money at $1:$1 that I could arrange that. Or I'll give emails and a warm intro for free.
FYI @jefftk, the link from your site for this post goes to https://www.lesswrong.com/posts/slug/qNJnXBFzninFT5m3nkids which is a 404 for me
Seeing you know the exact numbers, I wonder if you could connect with those other families? It's harder to get the best outcome if players do not cooperate and do not even know the wishes of others. Adding that this would be a valuable socializing opportunity would be somewhat hypocritical from me but it's still so.
Our oldest is finishing up 5th grade, at the only school in our city that doesn't continue past 5th. The 39 5th graders will be split up among six schools, and we recently went though the process of indicating our preferences and seeing where we ended up. The process isn't terrible, but it could be modified to stop giving an advantage to parents who carefully game it out while better matching kids to preferred schools.
First, what is the current process? You put in 1st, 2nd, and 3rd choice rankings, which are interpreted in three rounds. Kids are assigned to 1st choice schools, then the ones who didn't get in are assigned among 2nd choices, and finally 3rd choices. Ties are broken by sibling priority, proximity priority, and then by lottery number.
For sibling priority, if you have a sibling who will be in the school next year you have priority over students who don't. In practice this means if you list a sibling priority school as your first choice you get it.
For proximity priority, each family has a proximity school. This may not be the closest one to their house, and for us it isn't, but its at least reasonably close. It's the same as siblings: you have priority over any non-proximity students. Listing your proximity school first won't always get you in, since some schools (ex: ours) have many more proximity students than open spots.
The open spots this year are:
| School | Sibling | Proximity | Available seats |
|---|---|---|---|
| A | 0 | 3 | 2 |
| B | 0 | 3 | 0 |
| C | 0 | 3 | 10 |
| D | 2 | 11 | 23 |
| E | 1 | 16 | 4 |
| F | 0 | 0 | 4 |
Under the current system, what did it make sense for us to put for our top three choices? Ignoring B, which has no available spots, our preference order is D > E > A > C > F. We could put that down directly (D, E, A) but how do proximity and limited spaces affect our decision?
Our proximity school is E, with 4 available seats. It was very likely that the family with sibling priority would put it first, so really 3 available seats. If we put it first and so did all other families with proximity, we'd have a 3/15 chance of getting a spot there. I think this means our best chances would be putting first D, then C, and then it doesn't matter much:
While we have proximity at E, since there are so many more E-proximal students than spots, even if it was our top choice I'd only put it first if we thought "E vs everything else" was the key question. But since we prefer D, and since I expect enough proximity students will put E first that it will go in the first round, we shouldn't list it at all: that would waste our 2nd or 3rd pick.
Similarly, I expect A to go entirely to students with proximity, so no point listing it.
Putting our 1st choice on D makes sense to me: it's our actual first choice, and even after accounting for sibling and proximity students it still has ten open spots.
Then we should put C next, since we prefer it to F.
For simplicity, lets assume everyone has the same preferences we would have if we lived where they did. That means people prefer whichever is closest of A, E, or D. Then on the first round, of the 39 rising 6th graders:
So our odds of getting D would be somewhere between 10/20 and 10/15.
But the real world looks a bit better than this:
Some kids are probably moving out of district, though they may wait until after they know their school assignment to decide.
Not every family has the same preferences.
Some families don't game this out carefully. I especially think it's likely that too many families who are close to indifferent between D and E put E first on the basis of it being their proximity school.
When we put in our preferences I guessed our likely outcomes were 62% D, 35% C, 2% other. Several weeks later we learned that our lottery number was 19/39, we got C and were placed first on the waitlist for D. Since there are ~70 rising sixth graders for D I think it's very likely that at least one of them will move away and we'll get in.
This felt a bit like playing a board game because that's the main place I work through rules in a zero-sum context, but here the results matter. I really don't like that us getting a school we prefer essentially has to come at the expense of other families getting what they'd prefer.
While the zero-sum nature is unavoidable, we could at least rework the system to no longer require families to be strategic. This is actually a very well-known problem, and we can apply the Gale–Shapley algorithm, which is used in medical residency matching:
Instead of listing just your top three choices, you list all of them. Because there's no benefit to misreporting your preferences this is relatively easy. Once you have everyone's preferences you assign lottery numbers as before, and then run multiple rounds of an algorithm.
In the first round, every student "applies" to their top choice. The school ranks students by sibling status, then proximity status, then lottery number, and provisionally accepts students up to capacity. In the next round unassigned students "apply" to their next ranked schools, with schools provisionally accepting anyone they rank higher than their previously provisionally accepted students and bumping students as needed. This continues until everyone has a place, and which point provisional acceptances become real acceptances and students are notified.
I especially like that with this algorithm families don't need to consider what other families are likely to do. If they prefer E to D, they can just put E first, without worrying that they are wasting a choice. While as someone who does think through strategy I expect this change would make our family mildly worse off, a system where people have the best chances of getting into their preferred schools if they accurately report their preferences seems clearly better overall.