The access weight using C A(c) →
C(1-A(c))/[(1-C)+C(1-A(c))], where C is the
access fraction and A(c)=Pr(S>=c) the reported-score
tail. Nonreporting uses the below-cutoff population (lines 1134–1172,
1233–1267).
The current fixed-point proof explicitly assumes continuous
lower-tail mass and first moment, positive denominator, and endpoint
signs (lines 1268–1390). Whether the source conditions imply this
package, or another proof avoids it, is unresolved here.
Theorem 3.2:
policy scope and operational blankness
Randomized reported-score estimates → deterministic reported output
in the checked theorem; the common no-report/no-take law may remain
arbitrary with finite expectations (lines 199–218, 256–395, 455–461,
1614–1778). For each supplied equilibrium and fixed public-feature
fibre, latent-skill or observable fairness implies either zero reporter
mass or equality of the actual output kernel to the no-report law almost
everywhere under the score law. In the report-required schedule, replace
reporters by takers, the no-report law by the no-take law, and the score
law by the Gaussian skill law. This asserts operational equality on
attained inputs, not every off-path input.
Equal expected estimates do not identify output laws:
delta_0 and (delta_-1+delta_1)/2 have mean
zero. The checked Gaussian example makes reported laws depend on score
sign while keeping their means zero, and gives nonreporters and students
without access the equilibrium reporter-mixture law. Deterministic
reported output removes this ambiguity; the example does not establish
that determinism is the only sufficient restriction.
The summary’s “demographic” reference → “observable,” matching its
argument (lines 1770–1782).
Equilibrium
timing, population laws, and active branches
Theorem 3.1: source equilibrium → also exclude profitable
group entry after recalibrating the school’s estimates. The
alternative changes behavior on positive mass in a measurable
public-feature region, preserves score technology and behavior
elsewhere, recalibrates attained posteriors, and satisfies the stated
response and strict entry-gain comparisons. Optional entry is tested on
any such region; required entry starts where current taking is zero.
Deriving this restriction from Definition 1 remains a formalization gap.
Necessity is unknown: the source may imply it, or another proof may
avoid it.
Voluntary Section 4: source equilibrium → maximal
self-enforcing participation. No admissible candidate can
strictly enlarge the selected active set, up to null sets, on any
positive-mass measurable public-feature region. Candidates satisfy the
same posterior, response, and entry/exit conditions. The active set
consists of takers who report under optional reporting, and takers under
required reporting. This is maximality under inclusion, not a greatest
set. Selected profiles exist, but proving the paper’s conclusions for
equilibria without this selection remains a formalization gap; necessity
of the restriction is unknown.
Source access “preset and uncorrelated with skill and features”
(line 207) → independence from the entire skill/noise block in the
one-student law. Zero correlation alone does not give the
conditional-law identities used here.
Lemma 4.1 and
Proposition 4.3: calculations
Lemma 4.1’s cutoff calculation →
c=(qtilde-intercept)/slope on the positive-slope domain,
obtained by solving intercept+slope*c=qtilde (lines
2183–2220, 2262–2310).
Proposition 4.3’s conditional-to-marginal comparison → a direct
calculation of unconditional posterior-mean variance (lines 2495–2505).
For prior variance v>0 and independent signal precision
sum J, that variance is v-(1/v+J)^(-1). An
additional positive-precision score strictly increases it, proving that
the two marginal estimate laws differ on the current equilibrium
domain.