# 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, every student "applies" to their top choice. The school ranks students by sibling status, then proximity status, then lottery number, and provisional 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.
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