  This paper introduces a novel measurement of informational size to school
choice problems, which inherits its ideas from Mount and Reiter (1974). This
concept measures a matching mechanism's information size by counting the
maximal relevant preference and priority rankings to secure a certain pairwise
assignment of a student to a school across all possible matching problems. Our
analysis uncovers two key insights. First, the three prominent strategy-proof
matching mechanisms, the deferred acceptance (DA) mechanism, the top trading
cycles (TTC) mechanism, and the serial dictatorship (SD) mechanism, is
(strictly) less informative than the non-strategy-proof immediate acceptance
(IA) mechanism. This result highlights a previously omitted advantage of IA in
term of its information demand, which partially explain the its popularity in
real-world matching problems especially when acquiring information is both
pecuniarily and cognitively costly. Second, when the matching problem contains
at least four students, the TTC demands less information compared to the DA to
implement a desired allocation. The issue of comparison between TTC and DA has
puzzled researchers both in theory (Gonczarowski and Thomas, 2023) and in
experiment (Hakimov and Kubler, 2021). Our result responds to this issue from
an informational perspective: in experiments with relatively fewer students,
agents tend to prefer DA over TTC as DA requires fewer information to secure
one's allocation in all problems (Guillen and Veszteg, 2021), while the
opposite is true when the market size increases (Pais et al., 2011). Among
others, our informational size concept offers a new perspective to understand
the differences in auditability (Grigoryan and Moller, 2024), manipulation
vulnerability (Pathak and Sonmez, 2013), and privacy protection (Haupt and
Hitzig, 2022), among some commonly used matching mechanisms.
