School choice using deferred acceptance algorithms increases competition for selective schools, by Terrier, Pathak and Ren

 Here's a working paper from the LSE which concludes that making it safe for parents to truthfully report their preferences increases the competition for selective schools (called grammar schools, which prioritize students based on admission tests), with the unintended consequence of disadvantaging poorer families in England. The paper contains a good description of past and present school assignment regimes in England.

From immediate acceptance to deferred acceptance: effects on school admissions and achievement in England by Camille Terrier Parag A. Pathak and Kevin Ren,  Centre for Economic Performance Discussion Paper No.1815, November 2021

"Abstract: Countries and cities around the world increasingly rely on centralized systems to assign students to schools. Two algorithms, deferred acceptance (DA) and immediate acceptance (IA), are widespread. The latter is often criticized for harming disadvantaged families who fail to get access to popular schools. This paper investigates the effect of the national ban of the IA mechanism in England in 2008. Before the ban, 49 English local authorities used DA and 16 used IA. All IA local authorities switched to DA afterwards, giving rise to a cross-market difference-in-differences research design. Our results show that the elimination of IA reduces measures of school quality for low-SES students more than high-SES students. After the ban, low-SES students attend schools with lower value-added and more disadvantaged and low-achieving peers. This effect is primarily driven by a decrease in low-SES admissions at selective schools. Our findings point to an unintended consequence of the IA to DA transition: by encouraging high-SES parents to report their preferences truthfully, DA increases competition for top schools, which crowds out low-SES students."

And here are the paper's concluding sentences:

" In England, selective schools pick students based on test scores, which favors high-SES parents. After the transition to DA, high-SES parents enroll at these schools at higher rates. Selective admissions are widespread throughout education, so our results provide an important caution to equity rationales for DA over IA in settings where selective schools have large market share."