This incentivises people to put popular schools which they don't like lower on their list (3-5), so that if they miss their top school, they can get into a popular school and swap with people who are at their desired school. Which removes the property that honesty in school rankings is the best policy.
Yeah, but what if they do like the popular school as a backup? Wouldn't eliminating swaps lower the average rank of assigned schools, and thus decrease happiness?
Lower average rank of assigned schools might mean decreased happiness, but this system also increases fairness because those who game the system tend to be better educated/more privileged. Disallowing swaps forces people to rank schools honestly and prohibits such gaming of a system.
