  We study a game theoretic model of standardized testing for college
admissions. Students are of two types; High and Low. There is a college that
would like to admit the High type students. Students take a potentially costly
standardized exam which provides a noisy signal of their type.
  The students come from two populations, which are identical in talent (i.e.
the type distribution is the same), but differ in their access to resources:
the higher resourced population can at their option take the exam multiple
times, whereas the lower resourced population can only take the exam once. We
study two models of score reporting, which capture existing policies used by
colleges. The first policy (sometimes known as "super-scoring") allows students
to report the max of the scores they achieve. The other policy requires that
all scores be reported.
  We find in our model that requiring that all scores be reported results in
superior outcomes in equilibrium, both from the perspective of the college (the
admissions rule is more accurate), and from the perspective of equity across
populations: a student's probability of admission is independent of their
population, conditional on their type. In particular, the false positive rates
and false negative rates are identical in this setting, across the highly and
poorly resourced student populations. This is the case despite the fact that
the more highly resourced students can -- at their option -- either report a
more accurate signal of their type, or pool with the lower resourced population
under this policy.
