  We consider a many-to-one matching market where colleges share true
preferences over students but make decisions using only independent noisy
rankings. Each student has a true value $v$, but each college $c$ ranks the
student according to an independently drawn estimated value $v + X_c$ for
$X_c\sim \mathcal{D}.$ We ask a basic question about the resulting stable
matching: How noisy is the set of matched students? Two striking effects can
occur in large markets (i.e., with a continuum of students and a large number
of colleges). When $\mathcal{D}$ is light-tailed, noise is fully attenuated:
only the highest-value students are matched. When $\mathcal{D}$ is long-tailed,
noise is fully amplified: students are matched uniformly at random. These
results hold for any distribution of student preferences over colleges, and
extend to when only subsets of colleges agree on true student valuations
instead of the entire market. More broadly, our framework provides a tractable
approach to analyze implications of imperfect preference formation in large
markets.
