# Middle School Choice

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:

*   Two or three list A with priority, two get it and zero or one miss out
*   Zero list B
*   Zero list C
*   Thirteen list D with priority, and fifteen to twenty, including us, list without priority
*   Four to ~eight list E with priority, four get it, and zero to four miss out.
*   Zero list F

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](https://en.wikipedia.org/wiki/Stable_marriage_problem), and we can apply the [Gale–Shapley algorithm](https://en.wikipedia.org/wiki/Gale%E2%80%93Shapley_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](https://en.wikipedia.org/wiki/Strategyproofness) 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.