Here's a model suggesting that people for whom it is more costly to gather information about school quality will do less well in preference based school choice.
Inattention and Inequity in School Matching, by Stefan F. Bucher & Andrew Caplin, NBER WORKING PAPER 29586, DOI 10.3386/w29586
Abstract: The attractive properties of the Deferred Acceptance (DA) algorithm rest on the assumption of perfect information. Yet field studies of school matching show that information is imperfect, particularly for disadvantaged students. We model costly strategic learning when schools are ex ante symmetric, agree on their ranking of students, and learning is rationally inattentive. Our analytic solution quantifies how each student’s rank, learning costs and prior beliefs interact to determine their gross and net welfare as well as the extent and form of mistakes they make. In line with the evidence, we find that lower-ranked students are affected disproportionately more by information costs, generally suffering a larger welfare loss than higher-ranked students. Interactions between mechanism design, inattention and inequity are thus of first order importance.
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"The challenge faced by matching models with endogenous information is that students face three sources of uncertainty: signal-based, deriving from uncertainty about what information their learning strategy will produce; strategic, deriving from uncertainty about others’ submissions and thus the resulting matching outcome; and value-based, referring to the remaining uncertainty about the student’s valuation of their tch.
We introduce a tractable model of strategically rational inattention in a matching market that parsimoniously captures this complexity. To focus on the interplay with inequity we assume that schools agree on their ranking of students. For analytic tractability we assume that schools are ex ante symmetric (exchangeable) and that learning is rationally inattentive (Sims, 2003; Caplin and Dean, 2015; Matejka and McKay, 2015). While our symmetry assumption implies that schools are ex ante identical, it does not require that students’ valuations are independent across schools so that information on a school can update beliefs about others.
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"A central finding is that DA exacerbates inequity. Lower-ranked students attain a lower fraction of their net welfare surplus under full information than do higher-ranked students, even if they have the same costs of learning. This is because lower-ranked students face greater uncertainty about the outcome resulting from any submission, which disperses and often dilutes their incentive to acquire information.
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"The fact that lower-ranking students are more likely to be matched with a school further down their list results in very unequal learning incentives..."
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