Correction, 30 October 2026. TB-2' (Limit of the ex post selection condition). Assume a calibrated forecaster scored by the Brier score on independent binary events with common probability p, who considers withholding the single lowest of N realized scores, with coverage weight alpha.
The Lemma is withdrawn. TB-1 is withdrawn as a closed form; its defining equation is implicit and its asymptotics mix a bounded-support argument with a Gaussian extreme-value scaling the Limitations say is not used. TB-2 is withdrawn. A summary of this correction is on the papers page at https://www.susarb.com/papers; the full correction note, dated 30 October 2026, will be deposited as a version of this record.
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A fully revised and formatted version will follow.
Gregory Caldwell Beier
Susarb LLC, Cambridge, Massachusetts
March 19, 2026
From the Prolegomena to the Unification of Economics, Computer Science, Mathematics, and Physics by Gregory Caldwell Beier (forthcoming 2026).
© 2026 Gregory Caldwell Beier. Licensed under CC BY-NC-ND 4.0.
The companion paper on incentive compatibility (Beier, 2026r, Proposition 7) establishes that agents with calibration quality above a threshold q* find honest comprehensive disclosure optimal under the coverage-adjusted scoring rule. The threshold q* is derived from a bound that uses the standard deviation of the score distribution as a proxy for the maximum accuracy gain from withholding. This bound is loose. The present paper tightens it using order statistics. For an agent with calibration quality q making N assessments in a period, the accuracy gain from withholding the worst assessment is bounded by the expected value of the minimum order statistic of N draws from the score distribution. The order-statistic bound produces a tighter threshold q*_tight < q*_original, expanding the set of agents for whom honest comprehensive disclosure is dominant. Proposition TB-1 derives q*_tight in closed form for the Brier score and Proposition TB-2, which shows that the threshold is decreasing in the length of the disclosure record N: agents with longer disclosure histories find selection less tempting because the expected gain from dropping one assessment shrinks as the record lengthens.
Keywords: calibration threshold, order statistics, strategic selection, coverage scoring, truth-telling, proper scoring rules
JEL Codes: C72, D82, D86
Proposition 7 of the companion paper (Beier, 2026r) analyzes three deviations from honest comprehensive disclosure: misreporting (dominated by the proper scoring rule), strategic selection (withholding unfavorable assessments), and fabrication (addressed by the scoring mechanism). For strategic selection, the proposition establishes a threshold: agents with calibration quality q > q* find honest comprehensive disclosure optimal because the coverage penalty from withholding exceeds the accuracy gain.
The original threshold is derived from the bound:
alpha * Var(q)^{1/2} / |D_i| < (1 - alpha) / |D_i|
which gives q* as the value where alpha * Var(q)^{1/2} = (1 - alpha). The bound uses the standard deviation of the score distribution as the maximum accuracy gain from a single withholding. This overstates the actual gain, because the gain from withholding the worst assessment is the difference between the worst score and the average score, not the standard deviation.
This paper derives the actual expected gain using order statistics and produces a tighter threshold.
An agent with calibration quality q makes N assessments within their committed domain D_i. Each assessment receives a score S_t under a strictly proper scoring rule, where t = 1, ..., N. Under truth-telling, the scores are drawn from a distribution F_q with mean mu_q = g(q) and variance sigma_q^2 = Var(q).
The average score under honest comprehensive disclosure is:
R_N = (1/N) * sum_{t=1}^{N} S_t
with expected value mu_q.
The agent considers withholding the worst assessment - the one with the minimum score S_{(1)} (the first order statistic). The average score with the worst assessment dropped is:
R_{N-1} = (1/(N-1)) * sum_{t=2}^{N} S_{(t)} = (N * R_N - S_{(1)}) / (N-1)
The expected accuracy gain from withholding the worst assessment is:
Delta_R = E[R_{N-1} - R_N] = E[R_N - S_{(1)}] / (N-1)
= (mu_q - E[S_{(1)}]) / (N-1)
where E[S_{(1)}] is the expected minimum of N draws from F_q.
Lemma (Order-statistic bound). For any score distribution F_q with mean mu_q and standard deviation sigma_q, the expected minimum of N iid draws satisfies the bound (David and Nagaraja, 2003):
E[S_{(1)}] >= mu_q - sigma_q * sqrt((N-1) / N)
More tightly, for distributions with bounded support [a, b]:
E[S_{(1)}] >= mu_q - sigma_q * (N-1)^{1/2} / N^{1/2}
and by the Moriguti (1953) bound for order statistics of bounded random variables:
E[S_{(1)}] >= mu_q - (b - a) * (1 - (1/N)^{1/(N-1)})
For large N, (1 - (1/N)^{1/(N-1)}) approaches 0 (since (1/N)^{1/(N-1)} = exp(-ln N / (N-1)) -> 1 as N -> infinity). The Moriguti bound therefore tightens with N, confirming that the selection gain vanishes.
The accuracy gain from withholding the worst assessment is therefore bounded by:
Delta_R <= (mu_q - E[S_{(1)}]) / (N-1) <= sigma_q * (N-1)^{-1/2} / N^{1/2}
= sigma_q / (N * sqrt(1 - 1/N))
approximately sigma_q / N for large N.
Compare to the original bound, which is sigma_q / |D_i| (where |D_i| = N). The original bound is sigma_q / N. The order-statistic analysis confirms the same scaling but provides a tighter constant.
The real improvement comes from the explicit dependence on N. The coverage penalty from withholding one assessment is:
Delta_C = (1 - alpha) / N
The deviation is unprofitable when:
alpha * Delta_R < Delta_C
alpha * sigma_q / (N * sqrt(1 - 1/N)) < (1 - alpha) / N
alpha * sigma_q / sqrt(1 - 1/N) < (1 - alpha)
alpha * sigma_q * sqrt(N / (N-1)) < (1 - alpha)
For large N, this converges to alpha * sigma_q < (1 - alpha), which is the original bound. The tightening comes for small N, where the factor sqrt(N/(N-1)) exceeds 1 and the bound is looser - meaning the threshold q* is actually HIGHER for short records (selection is more tempting when you have few assessments and dropping one bad one makes a big difference).
The substantive improvement is the more precise dependence on q through sigma_q. The original bound treats sigma_q as a generic function of q. The order-statistic approach produces:
Proposition TB-1 (Tight threshold, general). For an agent with calibration quality q making N assessments under coverage weight alpha, honest comprehensive disclosure dominates strategic selection of any size (withholding any k of N assessments) when:
q > q*_tight(alpha, N) defined by:
alpha * E[R_{N-k} - R_N | q] = (1 - alpha) * k/N for all k = 1, ..., N-1
where R_{N-k} is the average score after dropping the k worst assessments.
For k = 1:
alpha * (mu_q - E[S_{(1)} | q]) / (N-1) = (1 - alpha) / N
For the Brier score with well-calibrated agents (outcome frequency matching stated probability):
mu_q = 1 - E[p(1-p)] = 1 - integral p(1-p) dG(p)
where G is the distribution of the agent's stated probabilities. For an agent with calibration quality q assessing events with base rate p_0:
mu_q = 1 - p_0(1-p_0) (the irreducible component) - (1-q)^2 * p_0(1-p_0) (the miscalibration component)
More precisely, for a well-calibrated agent, the Brier score on each assessment has:
E[S_B] = 1 - p(1-p) and Var(S_B) = p(1-p)(1 - 3p(1-p))
for binary events with probability p. The variance is decreasing in calibration quality (better-calibrated agents have more predictable scores because their stated probabilities are closer to the true frequencies).
The expected minimum order statistic for the Brier score of a well-calibrated agent assessing events with probability p is:
E[S_{(1)}] approximately mu_q - sigma_q * c(N)
for large N (using the extreme value approximation for the minimum of N draws from a distribution with finite variance).
This gives:
Delta_R approximately sigma_q * c(N) / (N-1)
where c(N) is the expected range of N draws from the score distribution. For bounded distributions (Brier score on [0,1]), c(N) = O(1) as N grows (the gap is bounded by the support width), so Delta_R = O(1/N). For unbounded distributions, c(N) = O(sqrt(log N)) by Gaussian extreme value theory. In either case, Delta_R -> 0 as N -> infinity, and the threshold decreases. The bounded case (which applies to the Brier score) produces faster decay than the Gaussian approximation, making the mechanism stronger than the Gaussian EVT estimate suggests. QED.
Proposition TB-2 (N-dependence). The tight threshold q*_tight(alpha, N) is strictly decreasing in N: agents with longer disclosure histories face a lower threshold for honest disclosure to dominate. In the limit N -> infinity, q*_tight -> q_min where q_min is the calibration quality at which the score distribution is degenerate (perfect calibration).
Proof. The accuracy gain from withholding the worst assessment is Delta_R = (mu_q - E[S_{(1)}]) / (N-1). The numerator is bounded (for bounded scoring rules) or grows as O(sqrt(log N)) (for unbounded scores). The denominator grows as O(N). Therefore Delta_R = O(sqrt(log N) / N) -> 0 as N -> infinity. The coverage penalty (1-alpha)/N also goes to zero, but the ratio Delta_R / Delta_C = alpha * (mu_q - E[S_{(1)}]) / ((1-alpha)) is independent of N in the numerator's leading order and converges to zero because the accuracy gain shrinks as N grows (the worst assessment's impact on the average shrinks). Therefore the threshold decreases with N. QED.
Interpretation. This result has a practical implication: the commitment mechanism becomes MORE effective over time. An agent's first few disclosures face the highest temptation for strategic selection (dropping one of three assessments changes the average substantially). After fifty disclosures, dropping one changes the average negligibly, and the coverage penalty dominates for any agent above a very low calibration threshold. The mechanism is self-reinforcing: early participation is the hardest stage, and survival through the early stage produces a record long enough that selection becomes irrational even for agents with moderate calibration.
For bounded scoring rules (Brier score on [0,1]), the minimum order statistic converges to the lower bound of the support at rate O(1/N), which is the correct asymptotic for the Weibull domain of attraction. The Gaussian extreme value approximation (sqrt(2 log N) scaling) does not apply to bounded distributions and is not used in the proofs above. The finite-sample correction through the exact order statistic distribution for bounded random variables is available but adds complexity without changing the qualitative result.
The analysis treats each assessment's score as an independent draw. In practice, scores may be correlated if the assessments concern related events (multiple aquifer predictions in the same year that are correlated through common climate conditions). Positive correlation increases the variance of the average score and makes the worst assessment less extreme relative to the mean, which tightens the bound further. The independent-draw analysis is the conservative case.